Subgeometrically ergodic autoregressions

TL;DR

This paper explores how subgeometric ergodicity can be used to analyze the stationarity and mixing properties of higher-order nonlinear autoregressions, especially those resembling unit root processes, with slower convergence rates.

Contribution

It extends the concept of subgeometric ergodicity from first-order to higher-order nonlinear autoregressions, providing new conditions for their ergodic behavior.

Findings

Higher-order nonlinear autoregressions are subgeometrically ergodic under certain conditions.

These models exhibit stationarity and β-mixing with subgeometrically decaying coefficients.

The results generalize previous first-order cases to more complex autoregressive models.

Abstract

In this paper we discuss how the notion of subgeometric ergodicity in Markov chain theory can be exploited to study stationarity and ergodicity of nonlinear time series models. Subgeometric ergodicity means that the transition probability measures converge to the stationary measure at a rate slower than geometric. Specifically, we consider suitably defined higher-order nonlinear autoregressions that behave similarly to a unit root process for large values of the observed series but we place almost no restrictions on their dynamics for moderate values of the observed series. Results on the subgeometric ergodicity of nonlinear autoregressions have previously appeared only in the first-order case. We provide an extension to the higher-order case and show that the autoregressions we consider are, under appropriate conditions, subgeometrically ergodic. As useful implications we also obtain stationarity and $\beta$-mixing with subgeometrically decaying mixing coefficients.

Equations

$$\underset{n\to \infty }{\lim }{r}^n\Vert P^n(x;\cdot )-\pi {\Vert }_{TV}=0$$

$$\underset{n\to \infty }{\lim }r(n)\Vert P^n(x;\cdot )-\pi {\Vert }_{TV}=0.$$

$$y_t=g(y_{t-1})+{\epsilon }_t,t=1,2,\dots ,$$

$$\left|g(x)\right|\le \left(1-r{\left|x\right|}^{-\rho }\right)\left|x\right|\text{for }\left|x\right|\ge M_0[r>0,M_0>r^{1/\rho },0<\rho \le 2],$$

$$y_t=\left(1-\frac{r_0}{1+{\left|y_{t-1}\right|}^{\rho }}\right)y_{t-1}+{\epsilon }_t[r_0>0,0<\rho \le 2].$$

$$y_t=y_{t-1}+\tilde{g}(y_{t-1})+{\epsilon }_t,$$

$${\Vert \mu \Vert }_f=\underset{f_0:|f_0|\le f}{\sup }\left|\mu (f_0)\right|,$$

$$\lim_{n\to\infty}r(n)\lVert P^{n}(x\,;\,\cdot)-\pi\rVert_{f}=0\qquad\text{for }\pi\text{-almost all }x\in\mathsf{X}.\leavevmode\nobreak\$$

$$\underset{n\to \infty }{\lim }r(n)\underset{f_0:|f_0|\le f}{\sup }\left|E[f_0(X_n)\mid X_0=x]-\pi (f_0)\right|=0\text{for }\pi \text{-almost all }x\in \mathsf{X}$$

$$0<\underset{n\to \infty }{\mathrm{liminf}}\frac{r(n)}{r_0(n)}\le \underset{n\to \infty }{\mathrm{limsup}}\frac{r(n)}{r_0(n)}<\infty .$$

$$r_0(n)=(1+\ln (n){)}^{\alpha }\cdot (1+n{)}^{\beta }\cdot e^{c{n}^{\gamma }},\alpha ,\beta ,c\ge 0,\gamma \in (0,1).$$

$$E\left[V(X_1)|X_0=x\right]\le V(x)-\varphi \left(V(x)\right)+b{1}_C(x),x\in \mathsf{X}.$$

$$y_t={\phi }_1{y}_{t-1}+\cdots +{\phi }_p{y}_{t-p}+\tilde{g}(y_{t-1},\dots ,y_{t-p})+{\epsilon }_t,$$

$$\phi (z)=(1-z)(1-{\pi }_{1}z-\cdots -{\pi }_{p-1}{z}^{p-1}),$$

$$y_t=y_{t-1}+{\pi }_1\Delta y_{t-1}+\cdots +{\pi }_{p-1}\Delta y_{t-p+1}+\tilde{g}(y_{t-1},\dots ,y_{t-p})+{\epsilon }_t.$$

$$y_t={\pi }_1{y}_{t-1}+\cdots +{\pi }_{p-1}{y}_{t-p+1}+u_{t-1}+\tilde{g}(y_{t-1},\dots ,y_{t-p})+{\epsilon }_t$$

$$\left|u+\tilde{g}(\mathbf{x})-g(u)\right|\le \left|ϵ(\mathbf{x})\mathbf{x}\right|,$$

$$\left|g(u)\right|\le \{\begin{array}{ll}(1-r{\left|u\right|}^{-\rho })\left|u\right|& \text{for }\left|u\right|\ge M_0,\\ K_0& \text{for }\left|u\right|\le M_0.\end{array}$$

$$\left|u+\tilde{g}(\mathbf{x})\right|\le \left(1-r{\left|u\right|}^{-\rho }\right)\left|u\right|+o({\left|\mathbf{x}\right|}^{-d})\left|\mathbf{x}\right|\text{for }\left|u\right|\ge M_0.$$

$$\overline{g}(\mathbf{x})=x_1-{\pi }_1{x}_2-\cdots -{\pi }_{p-1}{x}_p+\tilde{g}(\mathbf{x})=u+\tilde{g}(\mathbf{x})$$

$$\left[\begin{array}[]{c}y_{t}\\ y_{t-1}\\ \vdots\\ \vdots\\ y_{t-p+1}\end{array}\right]=\begin{bmatrix}\pi_{1}&\pi_{2}&\cdots&\pi_{p-1}&0\\ 1&0&\cdots&0&0\\ 0&\ddots&\ddots&\vdots&\vdots\\ \vdots&\ddots&\ddots&0&0\\ 0&\cdots&0&1&0\end{bmatrix}\left[\begin{array}[]{c}y_{t-1}\\ y_{t-2}\\ \vdots\\ \vdots\\ y_{t-p}\end{array}\right]+\overline{g}(\boldsymbol{y}_{t-1})\left[\begin{array}[]{c}1\\ 0\\ \vdots\\ \vdots\\ 0\end{array}\right]+\varepsilon_{t}\left[\begin{array}[]{c}1\\ 0\\ \vdots\\ \vdots\\ 0\end{array}\right]$$

$${\mathbf{y}}_t=\mathbf{\Phi }{\mathbf{y}}_{t-1}+\overline{g}({\mathbf{y}}_{t-1}){\mathbf{\iota }}_p+{\epsilon }_t{\mathbf{\iota }}_p$$

$$\mathbf{A}=\left[\begin{array}{ccccc}1& -{\pi }_1& -{\pi }_2& \cdots & -{\pi }_{p-1}\\ 0& 1& 0& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& & \ddots & \ddots & 0\\ 0& \cdots & \cdots & 0& 1\end{array}\right]\text{and}\mathbf{\Pi }=\mathbf{A}\mathbf{\Phi }{\mathbf{A}}^{-1}=\left[\begin{array}{ccccc}0& 0& 0& \cdots & 0\\ 1& {\pi }_1& {\pi }_2& \cdots & {\pi }_{p-1}\\ 0& 1& 0& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & ⋮\\ 0& \cdots & 0& 1& 0\end{array}\right]=\left[\begin{array}{cc}0& {\text{0}}_{p-1}^{\prime }\\ {\mathbf{\iota }}_{p-1}& {\mathbf{\Pi }}_1\end{array}\right],$$

$$\mathbf{A}{\mathbf{y}}_t=\mathbf{\Pi A}{\mathbf{y}}_{t-1}+\overline{g}({\mathbf{y}}_{t-1}){\mathbf{\iota }}_p+{\epsilon }_t{\mathbf{\iota }}_p,$$

$$\left[\begin{array}{c}{z}_1({\mathbf{y}}_t)\\ {\mathbf{z}}_2({\mathbf{y}}_t)\end{array}\right]=\left[\begin{array}{cc}0& {\text{0}}_{p-1}^{\prime }\\ {\mathbf{\iota }}_{p-1}& {\mathbf{\Pi }}_1\end{array}\right]\left[\begin{array}{c}{z}_1({\mathbf{y}}_{t-1})\\ {\mathbf{z}}_2({\mathbf{y}}_{t-1})\end{array}\right]+\overline{g}({\mathbf{y}}_{t-1}){\mathbf{\iota }}_p+{\epsilon }_t{\mathbf{\iota }}_p=\left[\begin{array}{c}\overline{g}({\mathbf{y}}_{t-1})+{\epsilon }_t\\ {\mathbf{\Pi }}_1{\mathbf{z}}_2({\mathbf{y}}_{t-1})+z_1({\mathbf{y}}_{t-1}){\mathbf{\iota }}_{p-1}\end{array}\right].$$

$$V(\boldsymbol{x})=\tfrac{1}{2}\exp\bigl{\{}b_{1}\left|z_{1}(\boldsymbol{x})\right|^{b_{3}}\bigr{\}}+\tfrac{1}{2}\exp\bigl{\{}b_{2}\left\|\boldsymbol{z}_{2}(\boldsymbol{x})\right\|_{*}^{b_{3}}\bigr{\}}$$

$$V(\mathbf{x})=1+{\left|z_1(\mathbf{x})\right|}^{s_0}+s_1{\Vert {\mathbf{z}}_2(\mathbf{x})\Vert }_{\ast }^{\alpha s_0},$$

$$y_t-{\phi }_1{y}_{t-1}-\cdots -{\phi }_p{y}_{t-p}=\Delta y_t-{\pi }_1\Delta y_{t-1}-\cdots -{\pi }_{p-1}\Delta y_{t-p+1}=u_t-u_{t-1}=\tilde{g}(y_{t-1},\dots ,y_{t-p})+{\epsilon }_t,$$

$$u_t=u_{t-1}+I(u_{t-1})+{\epsilon }_t,$$

$$u_t-\nu =S(u_{t-1})(u_{t-1}-\nu )+{\epsilon }_t,$$

Full text

JEL classification: C22

Keywords: Nonlinear autoregressive model, subgeometric ergodicity, Markov chain, $\beta$-mixing.

1 Introduction

Markov chain theory and the notion of geometric ergodicity have become standard tools in econometrics and statistics when analyzing the stationarity and ergodicity of nonlinear autoregressions or other nonlinear time series models. A detailed discussion of the relevant Markov chain theory will be given in Section 2. For now, consider a Markov chain $X_{t}$ ($t=0,1,2,\ldots$) on the state space $\mathsf{X}$ and initialized from $X_{0}$ following some initial distribution (that is not necessarily the stationary distribution). Geometric ergodicity of $X_{t}$ entails that the $n$-step probability measures $P^{n}(x\,;\,\cdot)=\Pr\left(X_{n}\in\cdot\,|\,X_{0}=x\right)$ converge in total variation norm $\left\|\cdot\right\|_{TV}$ to the stationary probability measure $\pi$ at rate $r^{n}$ (for some $r>1$), that is,

[TABLE]

(the definition of $\left\|\cdot\right\|_{TV}$ and a formulation of (1) using a more general norm are given in Section 2). A common and convenient way to establish geometric ergodicity involves the verification of a so-called drift condition. Useful implications obtained with this approach include the existence of a stationary probability distribution $\pi$ of $X_{t}$ as well as the geometric $\beta$-mixing of $X_{t}$. (For a definition of $\beta$-mixing, see Doukhan (1994, Sec 1.1) or Bradley (2007, Ch 3).) The authoritative and classic reference to Markov chain theory is the monograph of Meyn and Tweedie (1993, 2009). Recent papers establishing geometric ergodicity of different nonlinear time series models include Francq and Zakoïan (2006), Ling (2007), Meitz and Saikkonen (2008), and Fokianos, Rahbek, and Tjøstheim (2009), among others.

In this paper we consider autoregressions that may exhibit rather arbitrary (stationary, unit root, explosive, nonlinear, etc.) behavior for moderate values of the observed series and that behave similarly to a unit root process for large values of the observed series. What this exactly means will be clarified shortly, but first we would like to emphasize that the autoregressions we consider will not necessarily be geometrically ergodic. Under appropriate conditions they will, nevertheless, satisfy a weaker form of so-called subgeometric ergodicity. A Markov chain is said to be subgeometrically ergodic when the convergence in (1) takes place at a rate $r(n)$ slower than geometric, that is,

[TABLE]

In the geometric case $r(n)=r^{n}$ with $r>1$ or, equivalently, $r(n)=e^{cn}$ with $c>0$. Examples of rates slower than geometric include subexponential rates (say, $r(n)=e^{cn^{\gamma}}$ with $c>0$ and $\gamma\in(0,1)$) and polynomial rates (say, $r(n)=(1+n)^{\beta}$ with $\beta>0$). For an up-to-date treatment of subgeometric ergodicity we refer to Chapters 16 and 17 of Douc, Moulines, Priouret, and Soulier (2018) (further references will be given below).

As will be discussed in Section 2, subgeometric ergodicity can conveniently be established by verifying a suitably formulated drift condition and useful implications analogous to those in the case of geometric ergodicity again follow. In particular, the existence of a stationary probability distribution $\pi$ of $X_{t}$ as well as the finiteness of certain moments are obtained. Moreover, in a companion paper Meitz and Saikkonen (2019) we show that subgeometric ergodicity implies $\beta$-mixing with subgeometrically decaying mixing coefficients. Subgeometric ergodicity therefore allows one to use limit theorems developed for $\beta$-mixing processes.

The main aims of this paper are to establish subgeometric ergodicity of certain higher-order nonlinear autoregressions and to illustrate the potential of the concept of subgeometric ergodicity for nonlinear time series models. To facilitate discussion, first consider a simple special case at an informal level. Specifically, consider the univariate first-order nonlinear autoregressive model

[TABLE]

where the error term $\varepsilon_{t}$ is a sequence of independent and identically distributed (IID) zero-mean random variables and $g$ is a real-valued function. For now, assume that $g$ is such that

[TABLE]

and that $g(x)$ is bounded for $\left|x\right|\leq M_{0}$. A concrete example where (4) can be easily verified is

[TABLE]

The model defined in (5) can be thought of as first-order autoregression with a time-varying autoregressive coefficient. For large values of $\left|y_{t-1}\right|$ the autoregressive coefficient takes values that are close to one and the generation mechanism of $y_{t}$ is close to a random walk while for small values of $\left|y_{t-1}\right|$ the autoregressive coefficient is close to $1-r_{0}$ and, for $r_{0}$ not very close to zero, $y_{t}$ is generated from a less persistent stationary autoregressive process. Overall, the generation mechanism of $y_{t}$ fluctuates between these two borderline cases. Simulated examples in Section 5 demonstrate that processes of the type described in (5) can exhibit behavior close to a random walk for rather long times before returning to a less persistent regime.

When $\rho\geq 1$ the model defined by equation (5) can be viewed as a special case of the model

[TABLE]

where the function $\tilde{g}$ is bounded (but not constant).111This model belongs to a class of models referred to as “random-walk-type Markov chains” by Jarner and Tweedie (2003); see particularly equation (3) of their paper. In Section 3, a higher-order version of equation (6) (without assuming boundedness) is used as a starting point of the formulation of our general model. Our main results in Section 4 show that, depending on the assumptions made, either geometric, subexponential, or polynomial ergodicity is obtained.

The preceding discussion illustrates what kind of behavior the autoregressions we consider may exhibit for large values of the observed series. However, it should be emphasized that inequality (4) restricts the regression function $g$ only for large values of its argument. As long as the assumed boundedness condition imposed on the function $g$ is satisfied, no restrictions are required when the process evolves in the vicinity of the origin. Allowing unit root type behavior for large (absolute) values of the observed series is the main feature which distinguishes the models we consider from most previous nonlinear autoregressions where stationary behavior is related to large (absolute) values of the process.222For instance, Lu (1998), Gouriéroux and Robert (2006), and Bec et al. (2008), among others, establish geometric ergodicity (and thus the existence of a stationary distribution) for autoregressions whose behavior approaches stationarity when the process moves away from the origin while in the vicinity of the origin its behavior can be rather arbitrary.

Previously results on subgeometric ergodicity of nonlinear autoregressions have been obtained in the probability literature by Tuominen and Tweedie (1994), Veretennikov (2000), Fort and Moulines (2003), Douc et al. (2004), Klokov and Veretennikov (2004, 2005), and Klokov (2007), among others (further discussion on these and some related papers will be provided in Section 4). To our knowledge, all of the previous results concern only first-order models. We contribute to this literature by obtaining results for more general higher-order autoregressions. This is achieved using techniques similar to those in the aforementioned papers, especially in Fort and Moulines (2003) and Douc et al. (2004). Depending on the assumptions imposed on the moments of the error term, the resulting rate of ergodicity is either geometric or subexponential or polynomial.

The rest of the paper is organized as follows. Section 2 contains basic concepts of Markov chains and summarizes existing results on subgeometric ergodicity. Section 3 introduces the nonlinear autoregressive model considered and states the assumptions used to obtain the results of the paper. The main results on subexponential and polynomial ergodicity are given in Section 4. In Section 5 we provide examples of our general model. Section 6 concludes. All proofs are collected in an Appendix and a Supplementary Appendix.

Finally, a few notational conventions are given. The minimum (maximum) of the real numbers $x$ and $y$ is denoted by $x\wedge y$ ($x\lor y$), and $L$ and $\Delta$ signify the lag operator and the difference operator, respectively (so that $\Delta x{}_{t}=(1-L)x_{t}=x_{t}-x_{t-1}$). The notation $\boldsymbol{1}_{S}(x)$ is used for the indicator function which takes the value one when $x$ belongs to the set $S$ and zero elsewhere, and $\left|\cdot\right|$ is used for both an absolute value and Euclidean norm. Furthermore, $\text{{0}}{}_{k}$ denotes a $k\times 1$ vector of zeros and $\boldsymbol{\iota}_{k}=(1,0,\ldots,0)$ ($k\times 1$).

2 Markov chains and subgeometric ergodicity

In this section we discuss basic concepts of Markov chains needed to obtain our results. More comprehensive discussions can be found in Meyn and Tweedie (2009) and Douc et al. (2018). Let $X_{t}$ ($t=0,1,2,\ldots$) be a Markov chain on a general measurable state space $(\mathsf{X},\mathcal{B}(\mathsf{X}))$ (with $\mathcal{B}(\mathsf{X})$ the Borel $\sigma$-algebra) and let $P^{n}(x\,;\,A)=\Pr(X_{n}\in A\mid X_{0}=x)$ signify its $n$-step transition probability measure. As in Fort and Moulines (2003) and Douc et al. (2004) our goal is to establish the convergence — in a suitably defined norm and at rate $r(n)$ — of the $n$-step probability measures $P^{n}(x\,;\,\cdot)$ to the stationary distribution $\pi$. To this end, let $f:\mathsf{X}\rightarrow[1,\infty)$ be an arbitrary fixed measurable function and, for any signed measure $\mu$, define the $f$-norm $\left\|\mu\right\|_{f}$ as

[TABLE]

where $\mu(f_{0})=\int_{x\in\mathsf{X}}f_{0}(x)\mu(dx)$ (and the supremum in (7) runs over all measurable functions $f_{0}:\mathsf{X}\to\mathbb{R}$ such that $\left|f_{0}(x)\right|\leq f(x)$ for all $x\in\mathsf{X}$). When $f\equiv 1$, the $f$-norm $\left\|\mu\right\|_{f}$ reduces to the total variation norm $\left\|\mu\right\|_{TV}=\sup_{f_{0}:\left|f_{0}\right|\leq 1}\left|\mu(f_{0})\right|$ used in (1) and (2).

We aim to establish that the $n$-step probability measures $P^{n}(x\,;\,\cdot)$ converge in $f$-norm and at rate $r(n)$ to the stationary probability measure $\pi$ satisfying $\pi(f)<\infty$, that is, that

[TABLE]

If (8) holds we say that the Markov chain $X_{t}$ is ($f,r$)-ergodic; this implicitly entails the existence of $\pi$ as well as certain moments as $\pi(f)<\infty$. (For instance, if $\mathsf{X}=\mathbb{R}$ and $f(x)=1+x^{2}$, then ($f,r$)-ergodicity implies that the stationary distribution of $X_{t}$ has finite second moments.)

In the probability literature the preceding definition of ($f,r$)-ergodicity is standard. However, an equivalent and more transparent formulation is obtained by replacing equation (8) with

[TABLE]

(see Tuominen and Tweedie (1994, p. 776)). For instance, if $f(x)=1+\left|x\right|$ the above equation shows that, for almost any initial value $x,$ the conditional expectation $E[X_{n}\mid X_{0}=x]$ converges to $\int_{x\in\mathsf{X}}x\thinspace\pi(dx)$, the expectation of the stationary distribution of $X_{t}$, and the rate of the convergence is given by $r(n)$.

Most of the recent ergodicity results obtained for nonlinear autoregressions have established geometric ergodicity so that the rate of convergence in (8) is given by $r(n)=r^{n}$, $r>1$. The subgeometric rate functions we consider are defined as follows (cf., e.g., Nummelin and Tuominen (1983) and Douc et al. (2004)). Let $\Lambda_{0}$ be the set of positive nondecreasing functions $r_{0}\,:\,\mathbb{N}\rightarrow[1,\infty)$ such that $\ln[r_{0}(n)]/n$ decreases to zero as $n\rightarrow\infty$. The class of subgeometric rate functions, denoted by $\Lambda$, consists of positive functions $r\,:\,\mathbb{N}\rightarrow(0,\infty)$ for which there exists some $r_{0}\in\Lambda_{0}$ such that

[TABLE]

Typical examples are obtained of rate functions $r$ for which these inequalities hold with (for notational convenience, we set $\ln(0)=0$)

[TABLE]

The rate function $r_{0}(n)$ is called subexponential when $c>0$, polynomial when $c=0$ and $\beta>0$, and logarithmic when $\beta=c=0$ and $\alpha>0$. Douc et al. (2004, Sec 3.3) consider subexponential convergence rates whereas Fort and Moulines (2003, Sec 2.2) consider polynomial convergence rates in model (3) (see also the related references mentioned in these papers).

The proofs of our results make use of the following condition adapted from Douc et al. (2018, Defn 16.1.7).444A somewhat more general version which allows $V$ to be extended-real-valued (i.e., $V\,:\,\mathsf{X}\rightarrow[1,\infty]$) is given in Douc et al. (2004).

Condition D. There exist a measurable function $V\,:\,\mathsf{X}\rightarrow[1,\infty)$, a concave increasing continuously differentiable function $\phi\,:\,[1,\infty)\rightarrow(0,\infty)$, a measurable set $C$, and a finite constant $b$ such that

[TABLE]

Conditions of this kind are known as drift conditions; when $\phi(v)=\lambda v$ for some $\lambda>0$ the so-called Foster-Lyapunov drift condition used to establish geometric ergodicity is obtained. For ease of discussion and reference, the following theorem summarizes geometric, subexponential, and polynomial ergodicity results that can be obtained using Condition D. (For the definitions of irreducibility, aperiodicity, and petite sets appearing in the theorem we refer the reader to Meyn and Tweedie (2009).)

Theorem 1** **((Meyn and

Tweedie (2009), Douc et al. (2004))).

Suppose $X_{t}$ is a $\psi$-irreducible and aperiodic Markov chain on $(\mathsf{X},\mathcal{B}(\mathsf{X}))$ and that Condition D holds with a petite set $C$ such that $\sup_{x\in C}V(x)<\infty$ and the function $\phi$ being either

• (i; geometric case)

$\phi(v)=\lambda v$* for some $\lambda>0$,*

• (ii; subexponential case)

$\phi(v)=c(v+v_{0})/[\ln(v+v_{0})]^{\alpha}$* for some $c,\alpha,v_{0}>0$, or*

• (iii; polynomial case)

$\phi(v)=cv^{\alpha}$* for some $\alpha\in[0,1)$ and $c\in(0,1]$.*

Then $X_{t}$ is $(f,r)$-ergodic with either

• (i)

$f=V$* and $r(n)=r^{n}$ for some $r>1$ (or, equivalently, $r(n)=(e^{c})^{n}$ for some $c>0$),*

• (ii)

$f=V^{\delta}$* and $r(n)=(e^{d})^{n^{1/(1+\alpha)}}$ for any $\delta\in(0,\negthinspace 1)$ and any $d\in(0,(1\negthinspace-\negthinspace\delta)\negthinspace\left\{c(1\negthinspace+\negthinspace\alpha)\right\}^{1/(1+\alpha)})$, or*

• (iii)

$f=V^{1-\delta(1-\alpha)}$* and $r(n)=n^{\delta-1}$ for any $\delta\in[1,1/(1-\alpha)]$.*

In the geometric case the result of Theorem 1 is given in Meyn and Tweedie (2009), and in the subexponential and polynomial cases the result can be obtained from Douc et al. (2004); some further details are provided in the proof of Theorem 1 in the Supplementary Appendix. Note that in the subexponential case choosing $v_{0}$ sufficiently large ensures the concavity of $\phi$ required in Condition D and also that (in the subexponential case) results with a faster rate of convergence and/or larger $f$-norm could be obtained at the expense of more complex notation; see Douc et al. (2004, Sec 2.3).

An essential feature of the subgeometric ergodicity results in Theorem 1 is that there is a trade-off between the rate of convergence and the size of the $f$-norm; in Theorem 1 the choice of $\delta$ reflects this. If a fast rate of convergence is desired one has to accept a small $f$-norm (recall from (8) that the size of the $f$-norm is directly proportional to the order of finite moments the stationary distribution is guaranteed to have). For instance, in the polynomial case choosing $\delta=1/(1-\alpha)$ gives the fastest rate of convergence and with this choice the $f$-norm reduces to the total variation norm (so that $f\equiv 1$); the extreme case $\alpha=0$ results in $r(n)\equiv 1$ and standard ergodicity. In the subexponential case values of $\delta$ that are close to zero (one) correspond to small (large) $f$-norms.

It is also worth noting that Condition D is only sufficient, not necessary, for ($f,r$)-ergodicity. It is therefore possible that with another drift condition (not necessarily a special case of Condition D) a better rate function could be obtained, but presumably at the cost of a smaller norm. Being able to obtain necessary conditions for particular subgeometric ergodicity rates would be of interest but we will not pursue this issue. Necessary conditions for geometric and polynomial ergodicity in the context of random-walk-type Markov chains (see (6)) are given in Jarner and Tweedie (2003) (for an application of this result to a threshold autoregressive model, see Meitz and Saikkonen (2019)).

As already indicated in the Introduction, the ergodicity results of Theorem 1 imply results on $\beta$-mixing or, more specifically, on convergence rates of $\beta$-mixing coefficients $\beta(n)$ ($n=1,2,\ldots$) (for a definition of $\beta(n)$ and properties of $\beta$-mixing, see Doukhan (1994, Sec 1.1), Bradley (2007, Ch 3), or Meitz and Saikkonen (2019)). To illustrate this point, let $\mu$ signify the distribution of $X_{0}$, the initial value of the Markov chain $X_{t}$, and assume that$\int_{x\in\mathsf{X}}V(x)\mu(dx)<\infty$ (with $V$ as in Theorem 1). Then, using Theorems 1 and 2 of Meitz and Saikkonen (2019) the three cases in Theorem 1 imply the following convergence rates for $\beta$-mixing coefficients (here $c$ and $\alpha$ are as in Theorem 1):555See also the discussion following Theorem 2 of Meitz and Saikkonen (2019), and note that their Theorem 2(e) is also used to obtain the subexponential rate shown here.

• (i) geometric case:

$\lim_{n\rightarrow\infty}\tilde{r}^{n}\beta(n)=0$ for some $\tilde{r}>1$;

• (ii) subexponential case:

$\lim_{n\rightarrow\infty}(e^{\tilde{d}})^{n^{1/(1+\alpha)}}\beta(n)=0$ for any $\tilde{d}\in(0,\left\{c(1\negthinspace+\negthinspace\alpha)/2\right\}^{1/(1+\alpha)})$;

• (iii) polynomial case:

$\lim_{n\rightarrow\infty}n^{\alpha/(1-\alpha)}\beta(n)=0$.

Thus, the convergence rates of the $\beta$-mixing coefficients are qualitatively similar to the fastest convergence rates of ergodicity obtained in Theorem 1 (as indicated above, a slight improvement can be achieved in the subexponential case). These results, combined with the fact that the $(f,r)$-ergodicity given in Theorem 1 implies finiteness of moments, make possible to use limit theorems developed for $\beta$-mixing processes (and also for $\alpha$-mixing processes because $\beta$-mixing is known to imply $\alpha$-mixing).

3 Model and assumptions

We now introduce a higher-order generalization of the model discussed in the Introduction. Suppose the process $y_{t}$ ($t=1,2,\ldots$) is generated by

[TABLE]

where $\tilde{g}$ is a real-valued function, the error term $\varepsilon_{t}$ is a sequence of IID random variables, and exactly one of the roots of the polynomial $\varphi(z)=1-\varphi_{1}z-\cdots-\varphi_{p}z^{p}$ is equal to unity and (when $p\geq 2$) all others lie outside the unit circle. Thus, the regression function of the model has a linear part and a nonlinear part, and without the nonlinear part we have a standard linear $p$th order autoregression with a single unit root (cf. model (6)).

To express (10) in a different way, set $\pi_{j}=-\sum_{i=j+1}^{p}\varphi_{i}$ ($j=1,\ldots,p-1$; when $p=1$, set $\pi_{1}=\cdots=\pi_{p-1}=0$) so that we can express the polynomial $\varphi(z)$ as

[TABLE]

where the roots of the polynomial $\varpi(z)=1-\pi_{1}z-\cdots-\pi_{p-1}z^{p-1}$ lie outside the unit circle. This shows that we can write equation (10) alternatively as

[TABLE]

Denoting $u_{t}=y_{t}-\pi_{1}y_{t-1}-\cdots-\pi_{p-1}y_{t-p+1}$ equation (11) can be written as

[TABLE]

or as $u_{t}=u_{t-1}+\tilde{g}(y_{t-1},\ldots,y_{t-p})+\varepsilon_{t}$; when $p=1$ we obtain $y_{t}=y_{t-1}+\tilde{g}(y_{t-1})+\varepsilon_{t}$ as in (6). The formulation in (12) is convenient in our theoretical developments and will therefore be used instead of (11). One reason for this convenience is that in cases where the function $\tilde{g}$ depends on $y_{t-1},\ldots,y_{t-p}$ only through the linear combination $u_{t-1}=y_{t-1}-\pi_{1}y_{t-2}-\cdots-\pi_{p-1}y_{t-p}$ we can write equation (12) (with a slight abuse of notation) in a more compact way as $u_{t}=u_{t-1}+\tilde{g}(u_{t-1})+\varepsilon_{t}$. Then the process $u_{t}$ can be treated as the first-order model (6) and, as will be discussed shortly, with a suitable assumption, we can make use of results in Fort and Moulines (2003, Sec 2.2) and Douc et al. (2004, Sec 3.3) (this turns out to be the case even when $\tilde{g}$ is not a function of the process $u_{t-1}$ only).

Next we introduce the assumptions needed to prove our results. Our first assumption restricts the dynamics in equation (12).

Assumption 1**.**

Suppose the polynomial $\varpi(z)=1-\pi_{1}z-\cdots-\pi_{p-1}z^{p-1}$ and the function $\tilde{g}\,:\,\mathbb{R}^{p}\rightarrow\mathbb{R}$ in (12) satisfy the following conditions:

• (i)

The roots of $\varpi(z)$ lie outside the unit circle.

• (ii)

The function $\tilde{g}$ is measurable, bounded on compact subsets of $\mathbb{R}^{p}$, and there exists a measurable function $g\,:\,\mathbb{R}\rightarrow\mathbb{R}$ with the property $\left|g(x)\right|\rightarrow\infty$ as $\left|x\right|\rightarrow\infty$ such that the following two conditions hold.

• (ii.a)

With $\boldsymbol{x}=(x_{1},\ldots,x_{p})$ and $u=x_{1}-\pi_{1}x_{2}-\cdots-\pi_{p-1}x_{p}$, the function $\tilde{g}$ satisfies

[TABLE]

where $\epsilon(\boldsymbol{x})$ is a real-valued function such that $\left|\epsilon(\boldsymbol{x})\right|=o(\left|\boldsymbol{x}\right|^{-d})$ as $\left|\boldsymbol{x}\right|\rightarrow\infty$ for some $d>0$.

• (ii.b)

There exist positive constants $r$, $M_{0}$, $K_{0}$, and $0<\rho\leq 2$ such that for all $u\in\mathbb{R}$

[TABLE]

Assumption 1(i) corresponds to the conventional stationarity condition of a linear autoregression in that it requires the roots of the polynomial $\varpi(z)$ to lie outside the unit circle. In the first-order case $p=1$, this condition becomes redundant because then $\pi_{1}=\cdots=\pi_{p-1}=0$.

Assumption 1(ii) requires the function $\tilde{g}$ to be bounded on compact subsets and links it to another function $g$. Condition (ii.a) controls the difference between the functions $u+\tilde{g}(\boldsymbol{x})$ and $g(u)$ or, in model (12), the difference between the processes $u_{t-1}+\tilde{g}(y_{t-1},\ldots,y_{t-p})$ and $g(u_{t-1})$. In the special case where the function $\tilde{g}$ depends on $u$ only, condition (ii.a) becomes obvious because then one can choose $u+\tilde{g}(\boldsymbol{x})=g(u)$ and $\epsilon(\boldsymbol{x})=0$, and it suffices to check condition (ii.b) only. In this case we can use results in Fort and Moulines (2003, Sec 2.2) and Douc et al. (2004, Sec 3.3) directly in our proofs. However, we can do the same, albeit in a more complicated way, also when the function $\tilde{g}$ depends on the whole $p$-dimensional vector $\boldsymbol{x}$, but then the difference between the functions $u+\tilde{g}(\boldsymbol{x})$ and $g(u)$ may not increase “too fast” when $\left|\boldsymbol{x}\right|$ gets large. What is “too fast” is controlled by the function $\epsilon(\boldsymbol{x})$, and when $d\geq 1$ the difference between $u+\tilde{g}(\boldsymbol{x})$ and $g(u)$ becomes negligible when $\left|\boldsymbol{x}\right|$ increases.

Condition (ii.a) implies that $\left|u+\tilde{g}(\boldsymbol{x})\right|\leq\left|g(u)\right|+\left|\epsilon(\boldsymbol{x})\boldsymbol{x}\right|$, which combined with condition (ii.b) yields

[TABLE]

This fact is used in our proofs. Note also that condition (ii.a) is implied by the equality $u+\tilde{g}(\boldsymbol{x})=g(u)+\tilde{\epsilon}(\boldsymbol{x})\theta^{\prime}\boldsymbol{x}$ where $\theta$ is a $p$-dimensional parameter vector and, if $\tilde{\epsilon}(\boldsymbol{x})=o(\left|\boldsymbol{x}\right|^{-d})$ is assumed, condition (ii.a) holds with $\epsilon(\boldsymbol{x})=\left|\theta\right|\tilde{\epsilon}(\boldsymbol{x})$. This approach for checking condition (ii.a) is illustrated in Section 5.

Condition (ii.b) is similar to its first-order counterpart (4) to which it reduces when $p=1$. Note that apart from the boundedness condition, no restrictions are placed on $g(u)$ for moderate values of $u$. In the higher-order case this assumption concerns the filtered process $u_{t}=\varpi(L)y_{t}$. In the first-order case we also have $\boldsymbol{x}=u$ and the easiest way to verify Assumption 1(ii) may then be to define the function $g$ as $g(x)=x+\tilde{g}(x)$ (and $\epsilon(\boldsymbol{x})=0$), and verify condition (ii.b) directly. When $p\geq 2,$ the fact that the domain of the function $\tilde{g}$ is larger than that of $g$ complicates the situation in that then no simple connection between inequalities (13) and (14) can generally be found. An example of this case is provided in Section 5.

Our second assumption gives conditions required of the error term in equation (12).

Assumption 2**.**

$\{\varepsilon_{t},\,t=1,2,\ldots\}$* is a sequence of IID random variables that is independent of $(y_{0},\ldots,y_{-p+1})$ (with $p$ as in Assumption 1), the distribution of $\varepsilon_{1}$ has a (Lebesgue) density that is bounded away from zero on compact subsets of $\mathbb{R}$, and either*

• (a)

E\bigl{[}e^{\beta_{0}\left|\varepsilon_{1}\right|^{\kappa_{0}}}\bigr{]}<\infty* for some $\beta_{0}>0$ and $\kappa_{0}\in(0,1]$, and $E[\varepsilon_{1}]=0$; or*

• (b)

$E\left[\left|\varepsilon_{1}\right|^{s_{0}}\right]<\infty$* for some $s_{0}>\rho$ (with $\rho$ as in Assumption 1), and $E[\varepsilon_{1}]=0$ holds if $\rho\geq 1$.*

Assumption 2(a) corresponds to Assumption 3.3 of Douc et al. (2004, Sec 3.3), whereas Assumption 2(b) is a combination of the conditions imposed in (NSS 1), (NSS 4), and Lemma 3 of Fort and Moulines (2003, Sec 2.2). The boundedness condition imposed in Assumption 2 on the density of the error term is stronger than would be needed but is used for simplicity (see Assumption 3.3 of Douc et al. (2004, Sec 3.3) for a more general alternative).

Note that finiteness of the first expectation in Assumption 2(a) holds with $\kappa_{0}=1$ if the distribution of $\varepsilon_{1}$ has a moment generating function in some interval of the origin. Although many widely used distributions satisfy this condition some heavy tailed distributions are ruled out (this applies to distributions whose densities cannot be bounded by a term of the form $c_{1}e^{-c_{2}\left|x\right|}$ with $c_{1}$ and $c_{2}$ positive constants). An example is Student’s t-distribution irrespective of the value of the degrees of freedom parameter. The condition in Assumption 2(b) is used to address this issue. In this condition the case $0<\rho<s_{0}<1$ is rather extreme in that not even the expectation $E[\varepsilon_{1}]$ is assumed to exist.

4 Results

We now present our ergodicity results which we base on model (12). In Section 4.1 the rate of ergodicity established is subexponential whereas a slower polynomial rate of ergodicity is obtained in Section 4.2. The difference between these two cases stems from the assumed moment conditions: in Section 4.1 the condition in Assumption 2(a) is assumed whereas in Section 4.2 the weaker condition in Assumption 2(b) is employed. First we have to present the companion form of model (12) which applies to both of these cases and will be needed in the proofs of our theorems.

To simplify notation, denote $\boldsymbol{y}_{t}=(y_{t},\ldots,y_{t-p+1})$ and define the function $\overline{g}\,:\,\mathbb{R}^{p}\rightarrow\mathbb{R}$ as

[TABLE]

so that $\overline{g}(\boldsymbol{y}_{t-1})=u_{t-1}+\tilde{g}(\boldsymbol{y}_{t-1})$. It is readily seen that the companion form related to equation (12) reads as

[TABLE]

or, with obvious matrix notation,

[TABLE]

(when $p=1$, $\boldsymbol{\Phi}=0$). Thus, Assumption 2 ensures that $\boldsymbol{y}_{t}$ is a Markov chain on $\mathbb{R}^{p}$. For later purposes it is convenient to transform the companion form (16). To this end, we define the matrices

[TABLE]

where $\mathbf{A}$ is nonsingular and $\boldsymbol{\Pi}_{1}$ is the $(p-1)\times(p-1)$ dimensional lower right hand corner of $\mathbf{\Pi}$ (when $p=1$, $\mathbf{A}=1$ and $\mathbf{\Pi}=0$). With these definitions (16) can be transformed into

[TABLE]

where $\mathbf{A}\boldsymbol{y}_{t}=(u_{t},y_{t-1},\ldots,y_{t-p+1})$. Now, for any $p$-dimensional vector $\boldsymbol{x}$, form the partition $\boldsymbol{x}=(x_{1},\ldots,x_{p})=(x_{1},\boldsymbol{x}_{2})$ and define $\boldsymbol{z}(\boldsymbol{x})=(z_{1}(\boldsymbol{x}),\boldsymbol{z}_{2}(\boldsymbol{x}))=\boldsymbol{A}\boldsymbol{x}$, where (due to (17)) $z_{1}(\boldsymbol{x})=x_{1}-\pi_{1}x_{2}-\cdots-\pi_{p-1}x_{p}$ and $\boldsymbol{z}_{2}(\boldsymbol{x})=\boldsymbol{x}_{2}$ (when $p=1$, $\boldsymbol{x}_{2}$ and $\boldsymbol{z}_{2}(\boldsymbol{x})$ are dropped). With this notation equation (18) can be expressed as

[TABLE]

The first equation in (19) is now in a form that can be analyzed by using the results in Fort and Moulines (2003) and Douc et al. (2004). As for the second equation, by Assumption 1(i) the roots of the polynomial $\varpi(z)$ lie outside the unit circle, so that the eigenvalues of the matrix $\boldsymbol{\Pi}_{1}$ are smaller than one in absolute value. As is well known, this implies the existence of a matrix norm $\left\|\cdot\right\|_{*}$ induced by a vector norm, also denoted by $\left\|\cdot\right\|_{*}$, such that $\left\|\boldsymbol{\Pi}_{1}\right\|_{*}\leq\eta$ for some $\eta<1$ (see, e.g., Definition 5.6.1 and Lemma 5.6.10 in Horn and Johnson (2013)). These facts will be useful in our proofs.

4.1 Subexponential case

Our results make use of Condition D which requires choosing the function $V$. To this end, let $b_{1}$, $b_{2}$, and $b_{3}$ be positive constants whose values (to be specified later) depend on the constants $\beta_{0}$, $\kappa_{0}$, and $\rho$ introduced in Assumptions 1 and 2; for $b_{3}$ we already mention that it will always satisfy $b_{3}\in(0,1]$. When $p\geq 2$, we define the function $V$ as

[TABLE]

and when $p=1$, we define $V(x)=\exp\{b_{1}\left|x\right|^{b_{3}}\}$.

Now we can state the following theorem which makes use of the stronger moment requirement in Assumption 2(a). (The proof is given in the Supplementary Appendix.)

Theorem 2**.**

Suppose $p\geq 2$ and consider the Markov chain $\boldsymbol{y}_{t}$ defined in equation (16). Let Assumptions 1 and 2(a) hold, suppose that in Assumption 1 the constants $\rho$ and $d$ satisfy $0<\rho<2$ and $d=\rho/b_{3}$, and let $V(\boldsymbol{x})$ be as in (20).

• (i)

If $\rho>\kappa_{0}$, then $\boldsymbol{y}_{t}$ is ($f,r$)-ergodic with the subexponential convergence rate $r(n)=e^{kn^{b_{3}/\rho}}$ and the function $f$ given by $f(\boldsymbol{x})=V(\boldsymbol{x})^{\delta}$; this result holds for any choice of $\delta\in(0,1)$, for any $k$ such that $0<k<(1-\delta)\left\{c\rho/b_{3}\right\}^{b_{3}/\rho}$, and for some (small) $b_{1},\,b_{2}\in(0,\beta_{0})$, $b_{3}=\kappa_{0}\wedge(2-\rho)\in(0,1)$, and some (small) $c>0$.

• (ii)

If $\rho=\kappa_{0}$, then $\boldsymbol{y}_{t}$ is geometrically ergodic with the convergence rate $r(n)=e^{cn}$ and the function $f$ given by $f(\boldsymbol{x})=V(\boldsymbol{x})$; this result holds for some (small) $b_{1},\,b_{2}\in(0,\beta_{0})$, $b_{3}=\kappa_{0}\in(0,1]$, and some $c>0$.

When $p=1$, consider the Markov chain $y_{t}$ defined by $y_{t}=y_{t-1}+\tilde{g}(y_{t-1})+\varepsilon_{t}$. The above results hold for $y_{t}$ with the function $V$ defined as $V(x)=\exp\{b_{1}\left|x\right|^{b_{3}}\}$ (and the constant $d$ becomes redundant).

In Theorem 2, the case $\rho=\kappa_{0}$ represents a qualitative change in the ergodic behavior of the considered Markov chain: For $\rho>\kappa_{0}$ a slower subexponential convergence rate is obtained and $\rho=\kappa_{0}$ is the borderline case where a change to the faster geometric rate occurs. For $\rho<\kappa_{0}$, geometric ergodicity could also be established but we omit this case for brevity (in the first-order case this result is an immediate consequence of Theorem 3.3 of Douc et al. (2004, Sec 3.3)). Note also that, by the definition of the constant $b_{3}$, the rate of ergodicity in the subexponential case decreases as the value of $\rho$ increases.

Previously, Douc et al. (2004, Sec 3.3) obtained the results of Theorem 2 in the first-order case; our primary purpose here is to provide higher-order analogs of their results. Klokov and Veretennikov (2004, 2005) and Klokov (2007) have also studied the first-order model (3) satisfying inequality (4) with $1<\rho<2$ but otherwise their assumptions are rather different from ours. They obtain subexponential bounds for ergodicity in total variation norm (i.e., ($1,r$)-ergodicity) and for $\beta$-mixing coefficients but they do not discuss general ($f,r$)-ergodicity.

As discussed in Section 2, we can also establish $\beta$-mixing and, in contrast to Klokov and Veretennikov (2004, 2005) and Klokov (2007), we can permit all initial values with distribution $\mu$ such that $\int_{x\in\mathbb{R}^{p}}V(x)\mu(dx)<\infty$ (and $V$ as in Theorem 2). Specifically, the discussion at the end of Section 2 and Theorem 2 imply $\beta$-mixing with the following rates: In case (i) the rate is subexponential, i.e., $\lim_{n\rightarrow\infty}e^{\tilde{k}n^{b_{3}/\rho}}\beta(n)=0$ with any $\tilde{k}\in(0,\left\{c\rho/2b_{3}\right\}^{b_{3}/\rho})$, and in case (ii) the rate is geometric, i.e., $\lim_{n\rightarrow\infty}\tilde{r}^{n}\beta(n)=0$ for some $\tilde{r}>1$ (or, equivalently, $\lim_{n\rightarrow\infty}e^{\tilde{c}n}\beta(n)=0$ for some $\tilde{c}>0$).

The following corollary is an immediate consequence of the discussion after equality (8) (for a formal result, see Theorem 14.0.1 in Meyn and Tweedie (2009)).

Corollary** (to Theorem 2).**

Let $\pi$ signify the stationary distribution of $\boldsymbol{y}_{t}$ in Theorem 2 and let the function $f$ be as in cases (i) and (ii) of Theorem 2. Then $\pi(f)=\int_{\boldsymbol{x}\in\mathbb{R}^{p}}f(\boldsymbol{x})\pi(d\boldsymbol{x})<\infty$; in particular, $\pi(\left|\boldsymbol{x}\right|^{s})<\infty$ for all $s>0$ so that the stationary distribution has finite moments of all orders.

As we remarked after Theorem 1, in the subexponential case of Theorem 2 results with a faster rate of convergence and/or larger $f$-norm could be obtained at the expense of more complex notation. This means that in the above corollary finiteness of slightly larger moments could be obtained and the subexponential $\beta$-mixing rate discussed above could similarly be slightly improved.

4.2 Polynomial case

Next we consider ergodicity results relying only on the weaker moment requirement in Assumption 2(b). This will below lead to a slower polynomial rate of ergodicity. The key result used to relax the moment requirement is Lemma 3 of Fort and Moulines (2003, Sec 2.2) (which the authors use in conjunction with their analog of Condition D; we depart from their approach and use Condition D which corresponds to an analogous condition described in Section 1.2 in Fort and Moulines (2003)).

The function $V$ employed is now different from the subexponential case. When $p\geq 2$, we define the function $V$ as

[TABLE]

where $s_{0}$ is as in Assumption 2(b), $\alpha=1-\rho/s_{0}$ with $\rho$ as in Assumption 1(ii.b), and $s_{1}$ is a positive constant (to be specified later); when $p=1$, we define $V(x)=1+\left|x\right|^{s_{0}}$.

The following theorem presents the ergodicity result obtained when using the weaker moment condition in Assumption 2(b). (The proof is given in the Supplementary Appendix.)

Theorem 3**.**

Suppose $p\geq 2$ and consider the Markov chain $\boldsymbol{y}_{t}$ defined in equation (16). Let Assumptions 1 and 2(b) hold, suppose that in Assumption 1 the constants $\rho$ and $d$ satisfy $0<\rho\leq 2$ and $d=\rho/s_{0}$ when $s_{0}<1$ and $d=\rho$ when $s_{0}\geq 1$, and let $V(\boldsymbol{x})$ be as in (21). Assume further that either

• (i)

$0<\rho<1$* and $s_{0}>\rho$,*

• (ii)

$1\leq\rho<2$* and either $s_{0}=2$ or $s_{0}\geq 4$, or*

• (iii)

$\rho=2$* and $s_{0}\geq 4$ with $s_{0}r-\frac{1}{2}s_{0}(s_{0}-1)E[\varepsilon_{1}^{2}]>0$.*

Then $\boldsymbol{y}_{t}$ is ($f,r$)-ergodic with the polynomial convergence rate $r(n)=n^{\delta-1}$ and the function $f$ given by $f(\boldsymbol{x})=V(\boldsymbol{x})^{1-\delta\rho/s_{0}}$; this result holds for any choice of $\delta\in[1,s_{0}/\rho]$ and for some (small) $s_{1}>0$.

When $p=1$, consider the Markov chain $y_{t}$ defined by $y_{t}=y_{t-1}+\tilde{g}(y_{t-1})+\varepsilon_{t}$. The above results hold for $y_{t}$ with the functions $V$ and $f$ defined as $V(x)=1+\left|x\right|^{s_{0}}$ and $f(x)=1+\left|x\right|^{s_{0}-\delta\rho}$ (and the constant $d$ becomes redundant).

Options (i)–(iii) in Theorem 3 represent the combinations of the values of $\rho$ and $s_{0}$ for which the result can be obtained by relying on the corresponding cases (i)–(iii) in Lemma 3 of Fort and Moulines (2003, Sec 2.2). Unlike in Theorem 2, the case $\rho=2$ is allowed, but then an additional and rather intricate moment condition is required. A further departure from Theorem 2 is that the same polynomial rate of ergodicity is obtained in all cases. However, similarly to the subexponential case in Theorem 2, the rate of ergodicity decreases as the value of $\rho$ increases. Also, from the discussion at the end of Section 2 we can conclude that the rate of $\beta$-mixing implied by Theorem 3 is polynomial and, specifically, $\lim_{n\rightarrow\infty}n^{s_{0}/\rho-1}\beta(n)=0$.

The first-order case of Theorem 3 was obtained by Fort and Moulines (2003, Sec 2.2) (with slightly different assumptions). Polynomial ergodicity results for first-order autoregressions similar to that in (3) have previously appeared also in Tuominen and Tweedie (1994, Sec 5.2) (in the case $0<\rho<1$), Tanikawa (2001) (in the case $\rho=1$), and Veretennikov (2000) and Klokov (2007) (in the case $\rho=2$; these authors also obtain polynomial bounds for $\beta$-mixing coefficients but do not consider general ($f,r$)-ergodicity).

The following corollary on the moments of the stationary distribution is proved in the Supplementary Appendix. (In contrast to the subexponential case, using the ($f,r$)-ergodicity result of Theorem 3 would here yield a weaker moment result; hence, some extra steps are needed.)

Corollary** (to Theorem 3).**

Let $\pi$ signify the stationary distribution of $\boldsymbol{y}_{t}$ in Theorem 3. Then $\pi(f)=\int_{\boldsymbol{x}\in\mathbb{R}^{p}}f(\boldsymbol{x})\pi(d\boldsymbol{x})<\infty$ with $f(\boldsymbol{x})=\left|\boldsymbol{x}\right|^{s_{0}-\rho}$ so that the stationary distribution has finite moments up to order $s_{0}-\rho$.

5 Illustrative examples

In this section we discuss special cases of the general model introduced in Section 3. Using the three equivalent formulations (10)–(12) in Section 3, the model considered can be written as

[TABLE]

where the polynomial $\varphi(z)$ has precisely one unit root, can be decomposed as $\varphi(z)=(1-z)\varpi(z)$, and $u_{t}=\varpi(L)y_{t}$. Further formal assumptions will be stated in Propositions 1 and 2 below.

First-order subgeometrically ergodic autoregressions were already exemplified, albeit at a rather general level, in Fort and Moulines (2003, Sec 2.2) and Douc et al. (2004, Secs 3.3, 3.4). In Meitz and Saikkonen (2019, Sec 5) we study rates of subgeometric ergodicity and $\beta$-mixing in a first-order multi-regime self-exciting threshold autoregressive (SETAR) model; the proof of Theorem 3 in that paper illustrates how Theorems 2 and 3 of the present paper can be applied in a first-order case. In what follows we focus on examples of higher-order subgeometrically ergodic autoregressive models.

We consider three main examples. We first give a heuristic overview of them and then present the formalities. The first example we consider can be expressed as

[TABLE]

where $I(u_{t-1})$ is not constant and will be interpreted as a time-varying drift or intercept term (the case where $I(u_{t-1})$ is a constant is not of interest here because then model (22) reduces to a nonstationary unit root model (with or without a drift)). We consider the case where $I(u_{t-1})$ takes values in a bounded interval and fluctuates suitably between increasing and decreasing drifts. This ensures that model (22) will be (subgeometrically) ergodic and stationary under appropriate conditions (see Proposition 1 below) even though its dynamics involve a unit root component and a (time-varying) drift.

The second example we consider is

[TABLE]

where $\nu\in\mathbb{R}$ is an intercept term and $S(u_{t-1})$ will be interpreted as a time-varying slope coefficient. In the cases $S(u_{t-1})\equiv 0$ and $S(u_{t-1})\equiv 1$ model (23) reduces to the (linear) stationary model $\varpi(L)y_{t}=u_{t}=\varepsilon_{t}$ and to the nonstationary unit root model $\varphi(L)y_{t}=\varepsilon_{t}$, respectively. We consider the case of $S(u_{t-1})$ being time-varying, taking values in some interval $[s,1)$ ($s<1$), and attaining values arbitrarily close to 1 for values of $u_{t-1}$ large in absolute value. Under appropriate conditions (see Proposition 2 below), model (23) will be (subgeometrically) ergodic and stationary although exhibiting features similar to a unit root process.

Our third example generalizes the second one and illustrates that one can allow nonlinear dependence not only on $u_{t-1}$ but on the entire $\boldsymbol{y}_{t-1}$. Specifically, we consider model

[TABLE]

where we have omitted the intercept for simplicity, $S(u_{t-1})$ again represents a time-varying slope coefficient, and the term $F(\boldsymbol{y}_{t-1})$ captures the nonlinear dependence on the entire $\boldsymbol{y}_{t-1}$.

5.1 Example with time-varying intercept term of LSTAR type

We consider example (22) with the time-varying intercept term $I(u_{t-1})$ specified as in logistic smooth transition autoregressive (LSTAR) models (see, e.g., van Dijk et al. (2002)). Specifically, we choose

[TABLE]

with $L(u;b,a)=1/(1+e^{-b(u-a)})$ denoting the logistic function. The parameters $b,a_{1},a_{2}$ are assumed to satisfy $b>0$ and $a_{1}\leq a_{2}$ as usual, and $\nu_{1},\nu_{2}$ are assumed to satisfy $\nu_{1}<0<\nu_{2}$ to obtain ergodicity below. The time-varying intercept term $I(u_{t-1})$ now takes values in the interval $(\nu_{1},\nu_{2})$. Note that for large values of $b$ the logistic function $L(u;b,a)$ is close to the indicator function and then this model provides a close approximation to the above-mentioned threshold autoregressive model where $L(u_{t-1};b,a_{i})$ is replaced with an indicator function.

The following proposition shows the ergodicity of this model.

Proposition 1**.**

Consider the process $y_{t}$ defined by $u_{t}=u_{t-1}+I(u_{t-1})+\varepsilon_{t}$ as in (22) (with $u_{t}=\varpi(L)y_{t}$ and the roots of $\varpi(z)$ outside the unit circle), and with $I(u_{t-1})$ as in (25) (with $\nu_{1}<0<\nu_{2}$). Assume further that either

• (1)

Assumption 2(a) is satisfied with $\kappa_{0}\in(0,1)$,

• (2)

Assumption 2(a) is satisfied with $\kappa_{0}=1$, or

• (3)

Assumption 2(b) is satisfied with either $s_{0}=2$ or $s_{0}\geq 4$.

Then, under condition (1)/(2)/(3), the process $\boldsymbol{y}_{t}=(y_{t},\ldots,y_{t-p+1})$ is either

• (1)

subexponentially ergodic with convergence rate $r(n)=(e^{k})^{n^{\kappa_{0}}}$ (for some $k>0$),

• (2)

geometrically ergodic with convergence rate $r(n)=(e^{c})^{n}$ (for some $c>0$), or

• (3)

polynomially ergodic with convergence rate $r(n)=n^{s_{0}-1}$.

The proof of Proposition 1 is a straightforward application of Theorems 2 and 3 in the case $\rho=1$ (see the Appendix). Depending on moment assumptions the rate of ergodicity is geometric, subexponential, or polynomial, and similar rates also apply to $\beta$-mixing coefficients (see the discussions following Theorems 2 and 3 as well as the Corollaries to these theorems for existence of finite moments of the stationary distribution).

We now provide intuitive and graphical illustrations of the behavior processes covered by Proposition 1 can exhibit. An informal description captures the main features concisely: For ‘values of $u_{t-1}$ in the extreme left tail’, the intercept term $I(u_{t-1})$ is close to $\nu_{2}>0$ resulting in increasing drift towards ‘central values of $u_{t-1}$’; conversely, for ‘values of $u_{t-1}$ in the extreme right tail’, $I(u_{t-1})$ is close to $\nu_{1}<0$ and decreasing drift towards ‘central values of $u_{t-1}$’ takes place; finally, for such ‘central values of $u_{t-1}$’, unit root behavior without drift occurs. This informal description is illustrated in the top row of Figure 1 using three different cases of the function $I(\cdot)$ in (25); the precise parameter values used can be found in the caption of Figure 1.

The graph on the left illustrates the simplest possible case. When $u$ takes negative values the function $I(u)$ takes positive values; for $u<-2$ the values of $I(u)$ are very close to its supremum, in this example $0.08$. As $u_{t}=I(u_{t-1})+u_{t-1}+\varepsilon_{t}$, the behavior of the process $u_{t}$ is for $u_{t-1}<0$ (and particularly for $u_{t-1}<-2$) close to that of a unit root process with an increasing drift. A similar behavior occurs when $u_{t-1}$ takes positive values (and particularly when $u_{t-1}>2$) but then the drift is decreasing. Thus, large absolute values of $u_{t-1}$ induce a drift towards the origin which provides intuition why the process $u_{t}$ can be ergodic and stationary even though its behavior shows resemblance to a unit root process with a (negligible) drift near the origin where $I(u)\approx 0$. The other two graphs in the top row of Figure 1 illustrate similar but somewhat more involved possibilities for the function $I(u)$. The middle graph illustrates that the ‘unit root regime’ can take place over a wider range of values of $u$ and these values need not be centered at the origin; in this graph $I(u)\approx 0$ for $u$ roughly between $1.5$ and $3.5$. The graph on the right illustrates a possibility in which also the ‘middle regime’ corresponds to a unit root with positive drift.

The second row of Figure 1 presents three examples of simulated time series of $y_{t}$. These are generated from second-order versions of model (22) using the three time-varying intercept terms $I(u_{t-1})$ depicted in the first row. In all cases the autoregressive coefficient $\pi_{1}$ is equal to 0.75 and the error terms $\varepsilon_{t}$ are generated from rescaled Student’s $t$-distribution with five degrees of freedom (and different variances, see the caption for details). The third and fourth rows of Figure 1 show the time series graphs of the time-varying intercept terms $I(u_{t})$ and the autocorrelation functions of $y_{t}$, respectively, in these three examples. The time series graphs of $y_{t}$ in the second row bear a resemblance to those of unit root processes but nevertheless exhibit mean-reverting behavior. However, the mean reversion takes place slower than it would for a geometrically ergodic process. The autocorrelation functions also show very strong persistence. These features are related to the fact that the considered models are only polynomially ergodic with rate $r(n)=n^{4-\delta}$ for some $\delta>0$ (this follows from Proposition 1 as the error terms used only have moments of order smaller than five).

We notice from Figure 1 that the periods when $u_{t}$ takes large absolute values can be rather long and they can contain both increasing and decreasing periods for $y_{t}$. An example is the time series in the first column around $t\approx 800$ corresponding to the largest peak of the time series of $y_{t}$; similar features occur also in the time series in the second and third columns, often around peaks or troughs of the series. We also note from the second and third columns of Figure 1 that the time series of $I(u_{t})$ stays close to 0 or 0.03, respectively, for some time (corresponding to behavior of $u_{t}$ close to that of a unit root process without a drift or with a drift). An example is the rather long decreasing period in the time series of $y_{t}$ in the third column, roughly between $t\approx 200$ and $t\approx 500$, that is to large extent due to unit root type behavior.

5.2 Example with time-varying slope term of ESTAR type

Next we consider example (23) with the time-varying slope term $S(u_{t-1})$ being either

[TABLE]

where $r_{0}>0$ and the positive-valued function $h$ is such that $h(u)$ is large whenever $u$ is large in absolute value (formal requirements for $h$ are given in Proposition 2 below). Then the time-varying slope term $S(u_{t-1})$ takes values in some interval $[s,1)$ ($s<1$) and attains values arbitrarily close to 1 for values of $u_{t-1}$ large in absolute value. The shapes of $S_{1}(u)$ and $S_{2}(u)$ as functions of $u$ resemble the ‘inverted bell curve form’ commonly employed in exponential STAR (ESTAR) models (see, e.g., van Dijk et al. (2002)). The main difference between the two functions in (26) is that $S_{2}(u)$ takes values in the unit interval $(0,1)$ whereas $S_{1}(u)$ can also take negative values.

Before providing concrete examples we state the following proposition which imposes conditions on the function $h$ above to ensure that the results of Theorems 2 and 3 hold for model (23) with $S(\cdot)$ as in (26). The proof is straightforward and available in the Appendix.

Proposition 2**.**

Consider the process $y_{t}$ defined by $u_{t}-\nu=S(u_{t-1})(u_{t-1}-\nu)+\varepsilon_{t}$ as in (23) (with $u_{t}=\varpi(L)y_{t}$ and the roots of $\varpi(z)$ outside the unit circle) with $S(u_{t-1})$ being either $S_{1}(u_{t-1})$ or $S_{2}(u_{t-1})$ in (26) (with $r_{0}>0$) and with the function $h$ therein satisfying

• (h)

$h:\mathbb{R}\rightarrow(0,\infty)$* is measurable, bounded on compact sets, satisfies $h(u)\rightarrow\infty$ as $\left|u\right|\rightarrow\infty$, and is such that $c_{1}h(u)\leq\left|u\right|^{\rho}$ and $\left|u\right|^{\rho+c_{2}}\leq c_{3}h^{2}(u)$ for $\left|u\right|\geq M_{0}$, some $c_{1},c_{2},c_{3},M_{0}>0$, and $0<\rho\leq 2$.*

Assume further that either Assumption

• (1)

2(a) is satisfied with $\kappa_{0}<\rho$,

• (2)

2(a) is satisfied with $\kappa_{0}=\rho$, or

• (3)

2(b) is satisfied with an $s_{0}$ such that one of the conditions (i)–(iii) of Theorem 3 holds.

Then, under condition (1)/(2)/(3), the process $\boldsymbol{y}_{t}=(y_{t},\ldots,y_{t-p+1})$ is either

• (1)

subexponentially ergodic,

• (2)

geometrically ergodic, or

• (3)

polynomially ergodic.

Moreover, condition (h) above is satisfied for i) $h(u)=1+\lvert u-a\rvert^{\rho}$, ii) $h(u)=(1+\lvert u-a\rvert)^{\rho}$, iii) $h(u)=(1+(u-a)^{2})^{\rho/2}$, iv) $h(u)=1+\lvert u-a_{1}\rvert^{\rho_{1}}+\lvert u-a_{2}\rvert^{\rho_{2}}$, v) $h(u)=1+(1+\lvert u-a_{1}\rvert)^{\rho_{1}}+(1+\lvert u-a_{2}\rvert)^{\rho_{2}}$, or vi) $h(u)=1+(1+(u-a_{1})^{2})^{\rho_{1}/2}+(1+(u-a_{2})^{2})^{\rho_{2}/2}$ ($\rho,\rho_{1},\rho_{2}\in(0,2]$; $a,a_{1},a_{2}\in\mathbb{R}$).

The obtained rate of ergodicity is again either geometric, subexponential, or polynomial depending on the moment assumptions made (the precise convergence rates can be obtained from Theorems 2 and 3; for existence of moments, see the corollaries to Theorems 2 and 3). The last part of the proposition lists several potential concrete choices for the function $h$, of which cases (iii) and (vi) may be convenient if differentiability of the function $h$ is desired.

To illustrate the type of behavior processes covered by Proposition 2 may exhibit, consider as a simple example model (23) with $\nu=0$, the slope term $S_{1}(u_{t-1})$ in (26), and $h(u)=1+\lvert u\rvert^{\rho}$ (cf. (5)). The considered model then becomes

[TABLE]

The shape of the slope coefficient $S_{1}(u_{t-1})$ as a function of $u_{t-1}$ is now similar to that of an inverted bell curve increasing monotonically to unity as $\lvert u_{t-1}\rvert$ increases, and $S_{1}(u_{t-1})$ takes values within the interval $[1-r_{0},1)$. Note that depending on the value of $r_{0}$, the slope coefficient takes values potentially only near the unity (say, in the interval $[0.95,1)$) or even in rather extreme ranges (say, in the interval $[-100,1)$) so that very different behaviors can be accommodated.

To explain why such processes can still be ergodic and stationary, note that for large values of $\lvert u_{t-1}\rvert$ the slope $S_{1}(u_{t-1})$ always takes values within $(-1,1)$ (or a much smaller subset of it near unity). This prevents the process $u_{t}$ from exploding and ensures mean-reverting behavior. Note that for larger values of $\rho$ (within the permitted range $0<\rho\leq 2$), the slope $S_{1}(u_{t-1})$ approaches unity faster as $\lvert u_{t-1}\rvert$ increases, intuitively corresponding to more ‘wandering’ behavior of the observed series. This is reflected in Proposition 2 as slower rates of ergodicity being related to larger values of the parameter $\rho$ (see also Theorems 2 and 3).

Figure 2 illustrates the preceding discussion. The top row depicts examples of the function $S(u)$ with some particular choices of the function $h$ (see the caption of Figure 2 for the details). In the figures on the left and in the middle $S(u)=S_{1}(u)$ whereas in the figure on the right $S(u)=S_{2}(u)$. In each example the shape of the function $S(u)$ is similar to that of an inverted bell curve increasing monotonically to unity as $\lvert u\rvert$ increases.

Rows 2–4 of Figure 2 present simulated time series of $y_{t}$, time series graphs of the time-varying slope coefficients $S(u_{t-1})$, and the autocorrelation functions of $y_{t}$, respectively, in these three examples. In the first and second columns the time series graphs of $y_{t}$ and the related autocorrelation functions indicate unit root type behavior and strong persistence; in these cases also the slope coefficient $S(u_{t-1})$ takes values mostly larger than $0.90$ and often very close to one. Compared to these to cases, the time series graph of $y_{t}$ in the third column appears less ‘wandering’ and this feature is also reflected in the related autocorrelation function which decays faster and in a slope coefficient $S(u_{t-1})$ taking values further away from unity. Despite the three examples exhibiting somewhat different behaviours, all of them exhibit mean-reverting behavior and are subexponentially ergodic (this is due to Proposition 2 because $\rho>1$ and the error terms used are normally distributed and hence satisfy Assumption 2(a) with $\kappa_{0}=1$). The mean reversion again takes place slower than would be the case for geometrically ergodic processes.

5.3 Example of a more general formulation

Finally, we briefly consider example (24) and for simplicity set $p=2$. The time-varying slope term $S(u_{t-1})$ can be either one of the two options in (26). As for $F(\bm{y}_{t-1})$, we set $F(\bm{y}_{t-1})=\exp\{-\gamma\lvert\boldsymbol{y}_{t-1}\rvert^{2}\}(\theta_{1}y_{t-1}+\theta_{2}y_{t-2})$, where $\boldsymbol{y}_{t-1}=(y_{t-1},y_{t-2})$, $\gamma>0$, and $\theta=(\theta_{1},\theta_{2})$ can take any values in $\mathbb{R}^{2}$. That is, the considered model reads as

[TABLE]

On the right hand side of (28) the term $\exp\{-\gamma\lvert\boldsymbol{y}_{t-1}\rvert^{2}\}$ has a bell-shaped form (as a function of $\lvert\boldsymbol{y}_{t-1}\rvert$) with maximum at the origin while for choices such as $h(u_{t-1})=1+\lvert u\rvert^{\rho}$ the shape of $S(u_{t-1})$ is that of an inverted bell curve. Thus, given the shape of the terms $S(u_{t-1})$ and $\exp\{-\gamma\lvert\boldsymbol{y}_{t-1}\rvert^{2}\}$, model (28) can be viewed as a certain type of three-regime ESTAR model (see, e.g., van Dijk et al. (2002)).

It is straightforward to check that model (28) satisfies Assumption 1(ii) for both of the two options of $S$ and with $h$ as in Proposition 2 (details are available in the Appendix; note that in this example the function $\tilde{g}$ in equation (10) does not depend on $u_{t-1}$ only). It may be worth noting that Assumption 1(ii.a) does not hold if we replace the norm $\lvert\boldsymbol{y}_{t-1}\rvert$ in (28) with a linear function of $\boldsymbol{y}_{t-1}$ such as $u_{t-1}$. Another point worth noting is that Assumption 1(ii) does not rule out the possibility of setting $\pi_{1}=0$ (and $u_{t-1}=y_{t-1}$) in (28), and similarly for its higher-order counterparts where some or even all the coefficients $\pi_{1},\ldots,\pi_{p-1}$ may be equal to zero.

Allowing the parameters $\theta_{1}$ and $\theta_{2}$ in model (28) to be totally unrestricted highlights the fact that the autoregressions we consider may exhibit rather arbitrary behavior for moderate values of the observed series. As indicated in the Introduction, geometrically ergodic nonlinear autoregressions with features of this kind have previously been considered by Lu (1998), Gouriéroux and Robert (2006), Bec et al. (2008), and others. However, in most of these previous models stationarity is approached the further away the process moves from the origin whereas in our model unit root type behavior prevails for large absolute values of the process.

6 Conclusions

In this paper we examined the subgeometric ergodicity of certain higher-order nonlinear autoregressive models. Generalizing existing first-order results, we provided conditions that ensure subexponential and polynomial ergodicity of the considered autoregressions. These results were established by utilizing suitably formulated drift conditions. Relying on results in a companion paper Meitz and Saikkonen (2019), useful conclusions on the convergence rates of $\beta$-mixing coefficients were also obtained.

After obtaining theoretical results for rather general models we considered concrete examples and illustrated them with simulation. However, further work is needed to judge the usefulness of these models in practical applications. Several extensions could also be envisioned. For instance, subgeometric ergodicity of multivariate higher-order autoregressions or of models with conditional heteroskedasticity are interesting topics left for future work.

Appendix

This Appendix provides the technical details for the results in Section 5. Proofs of Theorems 1–3 and of Corollary to Theorem 3 are available in the Supplementary Appendix.

Proof of Proposition 1.

We first verify that Assumption 1 holds with $\rho=1$. Condition (i) holds by assumption. As $u_{t}=u_{t-1}+I(u_{t-1})+\varepsilon_{t}$, the function $\tilde{g}$ in condition (ii) depends on $u$ only and equals $I(u)$ (cf. eqn (12)) so that condition (ii.a) holds with $g(u)=u+I(u)$ and $\epsilon(\boldsymbol{x})=0$ (the condition $\left|g(u)\right|\rightarrow\infty$ as $\left|u\right|\rightarrow\infty$ clearly holds). Thus is suffices to check condition (ii.b) with $g(u)=u+I(u)$ and $I(u)$ as in (25). Clearly $g(u)$ is bounded on bounded subsets so the latter part of (14) holds. We now show that the former part of (14) holds with $\rho=1$, that is, that $\left|g(u)\right|\leq\left|u\right|-r$ for some $r,M_{0}>0$ and $\left|u\right|\geq M_{0}$. Note that the logistic function $L(u;b,a)$ satisfies $L(u;b,a)\to 0$ as $u\rightarrow-\infty$ and $L(u;b,a)\to 1$ as $u\rightarrow\infty$. As $I(u)=\nu_{1}L(u;b,a_{1})+\nu_{2}(1-L(u;b,a_{2}))$ with $\nu_{1}<0<\nu_{2}$, we can find a $M_{0}>0$ such that $I(u)>\nu_{2}/2>0$ for $u<-M_{0}$ and $I(u)<\nu_{1}/2<0$ for $u>M_{0}$. Setting $r=\min\{-\nu_{1}/2,\nu_{2}/2\}$ (and choosing a larger $M_{0}$ if necessary) we then have $\left|g(u)\right|=u+I(u)<u-(-\nu_{1}/2)\leq\left|u\right|-r$ for $u>M_{0}$ and $\left|g(u)\right|=-u-I(u)<-u-\nu_{2}/2\leq\left|u\right|-r$ for $u<-M_{0}$, establishing the former part of (14). Thus Assumption 1 holds with $\rho=1$.

To complete the proof, note that when condition (i) of the Proposition holds, Theorem 2(i) applies (with $b_{3}=\kappa_{0}$) and process $\boldsymbol{y}_{t}$ is subexponentially ergodic with convergence rate $r(n)=(e^{k})^{n^{\kappa_{0}}}$ (for some $k>0$). Similarly, when condition (ii) of the Proposition holds, Theorem 2(ii) applies and process $\boldsymbol{y}_{t}$ is geometrically ergodic with convergence rate $r(n)=(e^{c})^{n}$ (for some $c>0$). Finally, when condition (iii) of the Proposition holds, Theorem 3 applies and process $\boldsymbol{y}_{t}$ is polynomially ergodic with convergence rate $r(n)=n^{s_{0}-1}$. ∎

Proof of Proposition 2.

Assumption 1(i) again holds by assumption. For Assumption 1(ii), note that (23) can be written as $u_{t}=u_{t-1}+[1-S(u_{t-1})]\nu+[S(u_{t-1})-1]u_{t-1}+\varepsilon_{t}$ (cf. eqn (12)) and choosing $\tilde{g}(\boldsymbol{x})=[1-S(u)]\nu+[S(u)-1]u$ and $g(u)=[1-S(u)]\nu+S(u)u$ it is seen that Assumption 1(ii.a) holds with $\epsilon(\boldsymbol{x})=0$. For Assumption 1(ii.b), it again suffices to consider the former part of (14). The assumptions $c_{1}h(u)\leq\left|u\right|^{\rho}$ and $\left|u\right|^{\rho+c_{2}}\leq c_{3}h^{2}(u)$ made of function $h$ imply that, for large enough $\left|u\right|$,

[TABLE]

where (in the middle inequality) $1-\rho/2+c_{2}/2>0$ as $\rho\leq 2$ and $c_{2}>0$. Thus for the $S_{1}(u)$ in (26) and large enough $\left|u\right|$,

[TABLE]

where the upper bound is dominated by \bigl{(}1-\frac{c_{1}r_{0}/2}{\left|u\right|^{\rho}}\bigr{)}\left|u\right| for sufficiently large $\left|u\right|$ (so that Assumption 1(ii.b) holds with $r=c_{1}r_{0}/2$). For $S_{2}(u)=\exp\{-r_{0}/h(u)\}$ in (26), note that the inequality $1+x\leq e^{x}\leq 1+x+x^{2}/2$ (for $x\leq 0$) yields $1-S_{2}(u)\leq r_{0}/h(u)$ and $S_{2}(u)\leq 1-r_{0}/h(u)+r_{0}^{2}/2h^{2}(u)$ (for all $u$). These inequalities, together with the fact that $0<S_{2}(u)<1$, imply that

[TABLE]

Using the inequalities in (29) implies that, for large enough $\left|u\right|$, the above upper bound is dominated by

[TABLE]

which in turn is again dominated by \bigl{(}1-\frac{c_{1}r_{0}/2}{\left|u\right|^{\rho}}\bigr{)}\left|u\right| for sufficiently large $\left|u\right|$ (so that Assumption 1(ii.b) holds with $r=c_{1}r_{0}/2$).

The statements (1)–(3) now follow from Theorems 2(i), 2(ii), and 3, respectively.

Finally we show that condition (h) in the Proposition is satisfied for the six choices of $h(u)$ given. For case (i) note that, for $\left|u\right|>a$,

[TABLE]

where the product of the first two terms on the right hand side converges to one as $\left|u\right|\rightarrow\infty$, implying that, for some $c_{1}\in(0,1)$ and large enough $\left|u\right|$, $h(u)\leq\left|u\right|^{\rho}/c_{1}$ or, equivalently, $c_{1}h(u)\leq\left|u\right|^{\rho}$. Furthermore, let $c_{2}\in(0,\rho)$ and write $\lvert u\rvert^{2\rho}=\lvert u\rvert^{\rho-c_{2}}\lvert u\rvert^{\rho+c_{2}}$. Then the preceding discussion implies that, for some $c_{3}\in(0,1)$ and large enough $\left|u\right|$, $h^{2}(u)\geq\left|u\right|^{\rho+c_{2}}/c_{3}$ or, equivalently, $\left|u\right|^{\rho+c_{2}}\leq c_{3}h^{2}(u)$ . Thus, condition (h) holds in case (i).

Regarding cases (ii) and (iii), as $\rho\in(0,2]$, Loève’s $c_{r}$-inequality shows that $(1+\lvert u-a\rvert)^{\rho}\leq 2(1+\lvert u-a\rvert^{\rho})$, and the arguments used in case (i) above yield, for $\left|u\right|$ large enough, $(c_{1}/2)h(u)\leq\left|u\right|^{\rho}$ with $c_{1}$ as in case (i). Similarly in case (iii), $(1+(u_{t-1}-a)^{2})^{\rho/2}\leq 1+\lvert u_{t-1}-a\rvert^{\rho}$ so that $c_{1}h(u)\leq\left|u\right|^{\rho}$ holds with $c_{1}$ as in case (i). As for the inequality concerning $h^{2}(u)$, consider case (ii) and note that, for large enough $\left|u\right|$ and some $c_{2}\in(0,\rho)$,

[TABLE]

where the product of the first two terms on the right hand side tends to infinity as $\left|u\right|\rightarrow\infty$. Thus, as in case (i) we obtain $\left|u\right|^{\rho+c_{2}}\leq c_{3}h^{2}(u)$, and similar arguments can be used by replacing the definition of $h$ in the above equality with the one in case (iii).

Now consider case (iv) and let $\rho=\rho_{1}\lor\rho_{2}$. We then have, for large enough $\left|u\right|$, $h(u)\leq(1+\lvert u-a_{1}\rvert^{\rho})+(1+\lvert u-a_{2}\rvert^{\rho})$, and arguments used in case (i) above show that we can find $c_{1}\in(0,1)$ such that $c_{1}h(u)\leq\left|u\right|^{\rho}$ holds. Verifying the desired inequality for $h^{2}(u)$ can be established by using arguments used in case (ii) above. Cases (v) and (vi) can be handled with arguments similar to those already used; we omit the details. ∎

Checking Assumption 1(ii) for model (28)..

Suppose that $S(u_{t-1})$ is either one of the two options in (26) and that the function $h$ satisfies condition (h) in Proposition 2. Clearly, we can write model (28) as $u_{t}=u_{t-1}+[S(u_{t-1})-1]u_{t-1}+\exp\{-\gamma\lvert\boldsymbol{y}_{t-1}\rvert^{2}\}(\theta_{1}y_{t-1}+\theta_{2}y_{t-2})+\varepsilon_{t}$. Choosing $\tilde{g}(\boldsymbol{x})=[S(u)-1]u+\exp\{-\gamma\lvert\boldsymbol{x}\rvert^{2}\}\theta^{\prime}\boldsymbol{x}$ ($\boldsymbol{x}=(x_{1},x_{2})$) and $g(u)=S(u)u$ yields $u+\tilde{g}(\boldsymbol{x})-g(u)=\exp\{-\gamma\lvert\boldsymbol{x}\rvert^{2}\}\theta^{\prime}\boldsymbol{x}$, implying that Assumption 1(ii.a) holds with any positive $d$ (see inequality (15) and the following discussion). As $g(u)=S(u)u$, validity of Assumption 1(ii.b) can be checked as in the proof of Proposition 2 by setting the intercept term $\nu$ therein to zero. ∎

Supplementary Appendix to ‘Subgeometrically ergodic

autoregressions’ by Meitz and Saikkonen

Proof of Theorem 1.

In the geometric case, the result of Theorem 1 is given in Meyn and Tweedie (2009, Thm 15.0.1). In the polynomial case, the result can be obtained by combining Theorem 2.8 of Douc et al. (2004) with the discussion in their Section 2.3 (see also Jarner and Roberts (2002)). In the subexponential case, the function $\phi$ is concave and increasing as long as $v_{0}$ is chosen large enough (cf. Douc et al. (2008, p. 243, the paragraph following Assumption 2)). Again, the result can be obtained by combining Theorem 2.8 of Douc et al. (2004) with the discussion in their Section 2.3; note also that the two functions $\phi(v)=c(v+v_{0})/[\ln(v+v_{0})]^{\alpha}$ and $\phi_{0}(v)=cv/[\ln(v)]^{\alpha}$ both lead to the same rate function $r_{\phi}(n)$ given in Douc et al. (2004, p. 1365, line 6). ∎

Proof of Theorem 2. First note that from equation (16), Assumptions 1 and 2(a), and Theorem 2.2(ii) of Cline and Pu (1998) (see also Example 2.1 of that paper) it follows that the Markov chain $\boldsymbol{y}_{t}$ is a $\psi$-irreducible and aperiodic $T$-chain. Moreover, as in the proof of Lemma 1 of Lu (1998) it can be seen that $\psi$ is the Lebesque measure and using Theorem 6.2.5 of Meyn and Tweedie (2009) we can conclude that all compact sets of $\mathcal{B}(\mathbb{R}^{p})$ are petite (and in this case small, as shown by Theorem 5.5.7 of Meyn and Tweedie (2009)). The same also holds for the Markov chain $y_{t}$ in the case $p=1$.

In what follows, we first consider the case $p\geq 2$ and consider the case $p=1$ at the end of the proof.

Part (i): In this case we have $\rho>\kappa_{0}$ and $b_{3}=\kappa_{0}\land(2-\rho)\in(0,1)$; for brevity, the notation $b_{3}$ will be used. The choice of $b_{1}$ and $b_{2}$ will be discussed later. We can make use of results in the proof of Theorem 3.3, part (i), in Douc et al. (2004, Sec. 3.3). Write the function $V(\boldsymbol{x})$ as

[TABLE]

and consider $E\left[V(\boldsymbol{y}_{1})\mid\boldsymbol{y}_{0}=\boldsymbol{x}\right]$, the conditional expectation in (9). Note that $\boldsymbol{z}(\boldsymbol{y}_{1})$ appearing in $V(\boldsymbol{y}_{1})$ can be expressed as (see (19))

[TABLE]

In what follows, we usually drop the argument from $\boldsymbol{z}(\boldsymbol{x})$ and its components and write, for example, $\boldsymbol{z}_{2}$ instead of $\boldsymbol{z}_{2}(\boldsymbol{x})$. Now (dropping the argument from $\boldsymbol{z}(\boldsymbol{x})$)

[TABLE]

Defining $V_{\epsilon}(\boldsymbol{x})=\exp\{b_{1}\left|\epsilon(\boldsymbol{x})\boldsymbol{x}\right|^{b_{3}}\}$ we bound the expectation on the right hand side as follows:

[TABLE]

where the first inequality is due to the triangle inequality and the fact that $b_{3}\in(0,1)$, and the second inequality follows from the definition of the function $\overline{g}$ and inequality (13) in Assumption 1(ii). Thus, we can bound the conditional expectation $E\left[V(\boldsymbol{y}_{1})\mid\boldsymbol{y}_{0}=\boldsymbol{x}\right]$ as

[TABLE]

Step 1: Bounding $E\left[V_{1}\left(g(z_{1})+\varepsilon_{1}\right)\right]$ in (31). We first note that the arguments used by Douc et al. (2004) to obtain their inequality (3.14) can be used to justify that, for $\left|z_{1}\right|\geq M_{0}$,

[TABLE]

Moreover, repeating the arguments in Douc et al. (2004) between their (3.15)–(3.19) it can be shown that, for $\left|z_{1}\right|$ large (which, due to our Assumption 1(ii), also implies that $\left|g(z_{1})\right|$ is large) and some $c>0$,

[TABLE]

so that, for $\left|z_{1}\right|$ large,

[TABLE]

note that due to Assumption 2(a) and the choice of $b_{3}$, the condition E\bigl{[}\left|\varepsilon_{1}\right|^{2}V_{1}(\varepsilon_{1})\bigr{]}<\infty can be achieved by choosing the value of $b_{1}$ small enough. From the above inequalities it follows that, for $\left|z_{1}\right|$ large,

[TABLE]

where

[TABLE]

Next we obtain an upper bound for $k(z_{1})$. Note that we necessarily have $b_{3}-\rho<0$ and $2b_{3}-2\leq b_{3}-\rho$ with equality if and only if $b_{3}=2-\rho$ (these follow from $\rho>\kappa_{0}$ and $b_{3}=\kappa_{0}\land(2-\rho)$). First consider the case $2b_{3}-2=b_{3}-\rho$ so that $b_{3}=2-\rho$ and

[TABLE]

As $b_{3}-\rho<0$, the inequality $\frac{1}{2}b_{1}r^{2}b_{3}\left|z_{1}\right|^{b_{3}-\rho}\leq\frac{1}{2}\epsilon_{1}$ holds for all large enough $\left|z_{1}\right|$ and with $\epsilon_{1}>0$ which can be chosen as close to zero as desired. Moreover, as $b_{3}$ and $E\left[\varepsilon_{1}^{2}V_{1}(\varepsilon_{1})\right]$ are positive and here fixed, we can choose the value of $b_{1}$ small enough so that $b_{1}b_{3}E[\varepsilon_{1}^{2}V_{1}(\varepsilon_{1})]\leq\frac{1}{2}\epsilon_{1}$ holds. Hence, for $\left|z_{1}\right|$ large, $k(z_{1})\leq-\left[r-\tfrac{1}{2}\epsilon_{1}-\tfrac{1}{2}\epsilon_{1}\right]b_{1}b_{3}\left|z_{1}\right|^{b_{3}-\rho}$ and here $\epsilon_{1}$ can be chosen small enough so that $r-\epsilon_{1}>0$ holds. Now consider the case $2b_{3}-2<b_{3}-\rho$ (so that $b_{3}<2-\rho$). Write $k(z_{1})$ as

[TABLE]

and note that $\tfrac{1}{2}b_{1}r^{2}b_{3}\left|z_{1}\right|^{b_{3}-\rho}+b_{1}b_{3}E[\varepsilon_{1}^{2}V_{1}(\varepsilon_{1})]\left|z_{1}\right|^{(2b_{3}-2)-(b_{3}-\rho)}\leq\epsilon_{2}$ holds with $0<\epsilon_{2}<r$ for all large enough $\left|z_{1}\right|$ so that the bound $k(z_{1})\leq-(r-\epsilon_{2})b_{1}b_{3}\left|z_{1}\right|^{b_{3}-\rho}$ is obtained. To combine the two cases, note that the arguments above hold if $\epsilon_{1}$ and $\epsilon_{2}$ are replaced with $\epsilon_{3}=\epsilon_{1}\land\epsilon_{2}$. Thus, defining the positive constant $\omega_{1}$ as $\omega_{1}=r-\epsilon_{3}$ we obtain, for $\left|z_{1}\right|$ large,

[TABLE]

(cf. Douc et al. (2004, top of p. 1373)). Combining this with the inequality (32) we obtain

[TABLE]

Step 2: Bounding $V_{\epsilon}(\boldsymbol{x})E\left[V_{1}\left(g(z_{1})+\varepsilon_{1}\right)\right]$ in (31). Using the bound just obtained, bound the first term on the right hand side of (31) as

[TABLE]

For all $\left|z_{1}\right|$ large enough, $1-\omega_{1}b_{1}b_{3}\left|z_{1}\right|^{b_{3}-\rho}\in(0,1)$ and the same holds true for $k_{1}(z_{1})\,\overset{def}{=}\,1-\tfrac{1}{2}\omega_{1}b_{1}b_{3}\left|z_{1}\right|^{b_{3}-\rho}$. Using the inequality $(1-u)^{\alpha}\leq 1-\alpha u$ ($0\leq u,\alpha\leq 1$) we thus have

[TABLE]

Furthermore, as $\ln\left(k_{1}(z_{1})\right)=\ln(1-\tfrac{1}{2}\omega_{1}b_{1}b_{3}\left|z_{1}\right|^{b_{3}-\rho})\leq-\tfrac{1}{2}\omega_{1}b_{1}b_{3}\left|z_{1}\right|^{b_{3}-\rho}$ it follows that k_{1}(z_{1})=\exp\left\{\ln\left(k_{1}(z_{1})\right)\right\}\leq\exp\bigl{\{}-\tfrac{1}{2}\omega_{1}b_{1}b_{3}\left|z_{1}\right|^{b_{3}-\rho}\bigr{\}} and we can write

[TABLE]

Consider the argument of the exponential function on the right hand side of the above inequality. As $\boldsymbol{z}=(z_{1},\boldsymbol{z}_{2})=\boldsymbol{A}\boldsymbol{x}$, the equivalence of vector norms in $\mathbb{R}^{p}$ and straightforward calculations show that, for some $c_{*}>0$,

[TABLE]

where $\epsilon_{1}(\boldsymbol{x})=c_{*}\epsilon(\boldsymbol{x})$. Hence, as Assumption 1(ii) holds with $d=\rho/b_{3}$, we have $\left|\epsilon_{1}(\boldsymbol{x})\right|=o(\left|\boldsymbol{x}\right|^{-\rho/b_{3}})$ and

[TABLE]

so that, for all $\left|z_{1}\right|$ large (implying that $\left|\boldsymbol{z}\right|$ and hence that $\left|\boldsymbol{x}\right|$ is large666Due to the nonsingularity of the matrix $\boldsymbol{A}$, there exists a positive constant $c$ such that $c^{-1}\left|\boldsymbol{z}\right|\leq\left|\boldsymbol{x}\right|\leq c\left|\boldsymbol{z}\right|$, so that $\left|\boldsymbol{x}\right|\rightarrow\infty$ if and only if $\left|\boldsymbol{z}\right|\rightarrow\infty$.; see the discussion above Theorem 1),

[TABLE]

where $0<\omega_{2}<\tfrac{1}{2}\omega_{1}$. Thus, we can conclude that, for all $\left|z_{1}\right|$ large,

[TABLE]

Next define $\tau_{1}(z_{1})=1-\omega_{2}b_{3}\left|z_{1}\right|^{-\rho}$ and $\tau_{2}(z_{1})=1-\tau_{1}(z_{1})$, and note that $\tau_{1}(z_{1})\in(0,1)$ for any $\left|z_{1}\right|$ large. By the preceding discussion, we then have, for all $\left|z_{1}\right|$ large,

[TABLE]

Here the second inequality is justified by the convexity of the exponential function and the third one follows because $\tau_{1}(z_{1})\in(0,1)$ and $k_{1}(z_{1})\in(0,1)$ can be assumed. The last equality is due to the definition of $V_{1}$ and the definition of $\tau_{2}(z_{1})$ which implies

[TABLE]

where the inequality holds because $\left|z_{1}\right|\leq\left|\boldsymbol{z}\right|\leq c\left|\boldsymbol{x}\right|$ (see footnote 6) and where $o(1)\rightarrow 0$ as $\left|\boldsymbol{x}\right|\rightarrow\infty$.

It will be convenient to modify the preceding upper bound of $V_{\epsilon}(\boldsymbol{x})E\left[V_{1}\left(g(z_{1})+\varepsilon_{1}\right)\right]$. To this end, denote $\alpha=\rho/b_{3}-1$ ($>0$) and write b_{1}\left|z_{1}\right|^{b_{3}-\rho}=b_{1}^{\rho/b_{3}}\bigl{(}b_{1}\left|z_{1}\right|^{b_{3}}\bigr{)}^{-\alpha}\geq b_{1}^{\rho/b_{3}}\left(1+\ln V_{1}(z_{1})\right)^{-\alpha} where the inequality is based on the definition of $V_{1}(z_{1})$ (also note that $\ln(\frac{1}{2})\approx-0.6931$). Thus, by the definition of $k_{1}(z_{1})$ we have,

[TABLE]

Using this upper bound and the definition

[TABLE]

yields, for $\left|z_{1}\right|$ large and for a small enough choice of $b_{1}$, the following bound for the first term on the right hand side of (31):

[TABLE]

To state this more formally, we can find $b_{1}=\tilde{b}_{1}<\beta_{0}$, and $M_{1}\geq M_{0}$ such that the above inequality holds for $\left|z_{1}\right|>M_{1}$. Moreover, as in Douc et al. (2004, p. 1373) these choices can be done in such a way that, for some (finite) constant $\overline{M}_{1}$, and for all $z_{1}$,

[TABLE]

where $C_{1}=\left\{z_{1}\in\mathbb{R}\,:\,\left|z_{1}\right|\leq M_{1}\right\}$.

Step 3: Bounding $V_{2}\left(\boldsymbol{\Pi}_{1}\boldsymbol{z}_{2}+z_{1}\boldsymbol{\iota}_{p-1}\right)$ in (31). Here we assume that the choice of $b_{1}$ is fixed to the value $\tilde{b}_{1}$ specified above. Recall that V_{2}(\boldsymbol{\Pi}_{1}\boldsymbol{z}_{2}+z_{1}\boldsymbol{\iota}_{p-1})=\tfrac{1}{2}\exp\bigl{\{}b_{2}\left\|\boldsymbol{\Pi}_{1}\boldsymbol{z}_{2}+z_{1}\boldsymbol{\iota}_{p-1}\right\|_{*}^{b_{3}}\bigr{\}} and note that

[TABLE]

where we have made use of the fact $b_{3}\in(0,1)$ and Assumption 1(i) which implies that $\left\|\boldsymbol{\Pi}_{1}\right\|_{*}\leq\eta$ for some $\eta<1$ (see the discussion following equation (19)).

Let $\tau_{1}\in(0,1)$ and $\tau_{2}=1-\tau_{1}$ be such that $\tau_{2}\in(\eta^{b_{3}},1)$, and denote $b_{2,1}=b_{2}\left\|\boldsymbol{\iota}_{p-1}\right\|_{*}^{b_{3}}/\tau_{1}$ and $b_{2,2}=b_{2}/\tau_{2}$. Then,

[TABLE]

where the second inequality is justified by the convexity of the exponential function. Now, as $\tau_{2}\in(\eta^{b_{3}},1)$, we have $b_{2,2}\eta^{b_{3}}=b_{2}\eta^{b_{3}}/\tau_{2}<b_{2}$, and we choose the value of $b_{2}$ so small that $b_{2,1}=b_{2}\left\|\boldsymbol{\iota}_{p-1}\right\|_{*}^{b_{3}}/\tau_{1}<b_{1}=\tilde{b}_{1}$ with $\tilde{b}_{1}$ as fixed above.

We next bound $V_{2,1}(z_{1})$ and $V_{2,2}(\boldsymbol{z}_{2})$. For the former, write V_{2,1}(z_{1})=\exp\bigl{\{}-\left(b_{1}-b_{2,1}\right)\left|z_{1}\right|^{b_{3}}\bigr{\}}V_{1}(z_{1}) and use the facts $\ln V_{1}(z_{1})=\ln(\frac{1}{2})+b_{1}\left|z_{1}\right|^{b_{3}}$, $\alpha=\rho/b_{3}-1>0$, and $b_{1}-b_{2,1}>0$ to obtain

[TABLE]

where the inequality holds for any $\epsilon_{4}>0$ as long as $\left|z_{1}\right|$ is large enough. Using the definition of $\phi_{1}\left(V_{1}(z_{1})\right)$ in (34) this implies a bound for $-\phi_{1}\left(V_{1}(z_{1})\right)+V_{2,1}(z_{1})$ which will be needed later:

[TABLE]

where $\omega=\tfrac{1}{2}(\omega_{1}-\epsilon_{4})$ and, as $\omega_{1}>0$ holds for (fixed) $b_{1}=\tilde{b}_{1}$, we can choose $\epsilon_{4}$ so small that $\omega>0$ holds. Note that here the last expression provides a bound for $-\phi_{1}\left(V_{1}(z_{1})\right)+V_{2,1}(z_{1})$ that holds for all $z_{1}$ (although this may require redefining the set $C_{1}$ and the value of the constant $\overline{M}_{1}$ which appear also in the upper bound obtained earlier for $E\left[V_{1}\left(g(z_{1})+\varepsilon_{1}\right)\right]$). Denoting $\epsilon=\epsilon_{3}+\epsilon_{4}$ and using the definition of $\omega_{1}$ (given at the end of Step 1) we therefore have $\omega=\tfrac{1}{2}(r-\epsilon)$.

Now consider $V_{2,2}(\boldsymbol{z}_{2})=\frac{1}{2}\exp\{b_{2,2}\eta^{b_{3}}\left\|\boldsymbol{z}_{2}\right\|_{*}^{b_{3}}\}$ and recall that $b_{2,2}\eta^{b_{3}}<b_{2}$. Using the definition $V_{2}(\boldsymbol{z}_{2})=\frac{1}{2}\exp\{b_{2}\left\|\boldsymbol{z}_{2}\right\|_{*}^{b_{3}}\}$ we have, for some $\eta_{2}\in(0,1)$ and $\left\|\boldsymbol{z}_{2}\right\|_{*}$ bounded away from zero,

[TABLE]

and furthermore

[TABLE]

where the bound obtained above for $V_{2,1}(z_{1})$ has been omitted but it will be used below.

Step 4: Bounding $E\left[V(\boldsymbol{y}_{1})\,\left|\,\boldsymbol{y}_{0}=\boldsymbol{x}\right.\right]$ in (31). Using (35) and the preceding inequality obtained for V_{2}\bigl{(}\boldsymbol{\Pi}_{1}\boldsymbol{z}_{2}+z_{1}\boldsymbol{\iota}_{p-1}\bigr{)} we can now write

[TABLE]

As $\left|\boldsymbol{z}_{2}\right|\leq\left|\boldsymbol{z}\right|\leq c\left|\boldsymbol{x}\right|$ (see footnote 6), the term $o(1)$ on the right hand side converges to zero as $\left|\boldsymbol{z}_{2}\right|\rightarrow\infty$. Thus, as $V_{2}(\boldsymbol{z}_{2})=\frac{1}{2}\exp\{b_{2}\left\|\boldsymbol{z}_{2}\right\|_{*}^{b_{3}}\}$, we have, for $\left|\boldsymbol{z}_{2}\right|$ large,

[TABLE]

where $\eta_{3}\in(0,1)$. Hence,

[TABLE]

where $C_{2}=\left\{\boldsymbol{z}_{2}\in\mathbb{R}^{p-1}\,:\,\left|\boldsymbol{z}_{2}\right|\leq M_{2}\right\}$ and $M_{2}$ and $\overline{M}_{2}$ are some finite constants. Using this inequality and the bound in (36) we can bound $E\left[V(\boldsymbol{y}_{1})\,\left|\,\boldsymbol{y}_{0}=\boldsymbol{x}\right.\right]$ as follows:

[TABLE]

We still need to modify the right hand side of the above inequality to a form assumed in Condition D, and for simplicity we write this inequality as

[TABLE]

where $L\geq 2\overline{M}_{1}+\overline{M}_{2}$. Next note that $V(\boldsymbol{x})\geq V_{1}(z_{1})\geq 1/2$ (see (30)) so that

[TABLE]

Using these inequalities twice and defining $c_{\phi}=\omega b_{3}b_{1}^{\rho/b_{3}}\land\eta_{3}\left(1+\ln\frac{1}{2}\right)^{\alpha}$ ($>0$) we have

[TABLE]

Denoting $h(\boldsymbol{x})=c_{\phi}\left(1+\ln V(\boldsymbol{x})\right)^{-\alpha}$ we therefore obtain

[TABLE]

Because $V(\boldsymbol{x})\geq 1$ and $-\alpha<0$, we have $0<h(\boldsymbol{x})\leq c_{\phi}$ and $h(\boldsymbol{x})\rightarrow 0$, as $\left|\boldsymbol{x}\right|\rightarrow\infty$. Thus, for all $\left|\boldsymbol{x}\right|$ large enough, $h(\boldsymbol{x})\leq 1$, and therefore

[TABLE]

for all $\left|\boldsymbol{x}\right|$ large enough, where the first inequality is based on the inequality $(1-x)^{a}\leq 1-ax$ (which holds for $a,x\in[0,1]$) and the second inequality is justified by showing that the inequality

[TABLE]

holds for all $\left|\boldsymbol{x}\right|$ large enough. To show this, note first that

[TABLE]

so that it suffices to show that, for all $\left|\boldsymbol{x}\right|$ large enough, the right hand side of the last inequality is smaller than one or, equivalently, that $L<\tfrac{1}{2}h(\boldsymbol{x})\left(1-h(\boldsymbol{x})\right)^{\frac{1}{2}}V(\boldsymbol{x})$. This holds for all $\left|\boldsymbol{x}\right|$ large enough due to the definitions of $V(\boldsymbol{x})$ and $h(\boldsymbol{x})$ which imply that, as $\left|\boldsymbol{x}\right|\rightarrow\infty$, $V(\boldsymbol{x})\rightarrow\infty$ at an exponential rate (see (30)) whereas $h(\boldsymbol{x})\rightarrow 0$ at a logarithmic rate (see the above definition of $h(\boldsymbol{x})$).

We can therefore write inequality (37), for all $\left|\boldsymbol{x}\right|$ large enough, as

[TABLE]

As the right hand side is bounded when $\boldsymbol{x}$ belongs to any compact set, this further implies that there exist positive constants $M$ and $b$ such that for $C=\left\{\boldsymbol{x}\in\mathbb{R}^{p}\,:\,\left|\boldsymbol{x}\right|\leq M\right\}$ and for all $\boldsymbol{x}\in\mathbb{R}^{p}$

[TABLE]

where

[TABLE]

Now note that we can always find positive constants $v_{0}$ and $c$ such that the function $\phi(v)=c(v+v_{0})(\ln(v+v_{0}))^{-\alpha}$ is a concave increasing differentiable function for all $v\geq 1$ and such that

[TABLE]

for large enough $v$. Therefore, potentially redefining $M$, $b$, and $C$,

[TABLE]

Thus, we have verified Condition D (with $\alpha=\rho/b_{3}-1$). The result follows from Theorem 1.

Part (ii). Now $\rho=\kappa_{0}$ and, as in the proof of Theorem 3.3(ii) in Douc et al. (2004, p. 1373), many results in the proof of case $\rho>\kappa_{0}$ can be used. Again, we choose $b_{3}=\kappa_{0}\land(2-\rho)$, noting that now $b_{3}=\kappa_{0}$ and that the notation $\kappa_{0}$ will be used below instead of $b_{3}$. Also, the function $V(\boldsymbol{x})=V_{1}(z_{1})+V_{2}(\boldsymbol{z}_{2})$ is as in the case $\rho>\kappa_{0}$, and we need to bound the two terms in (31).

Step 1: Bounding $E\left[V_{1}\left(g(z_{1})+\varepsilon_{1}\right)\right]$ in (31). Exactly as in Part (i), Step 1, it again holds that, for $\left|z_{1}\right|>M_{0}$,

[TABLE]

and, for large $\left|z_{1}\right|$,

[TABLE]

Hence, for large $\left|z_{1}\right|$,

[TABLE]

where now

[TABLE]

Due to Assumption 2(a) and the choice of $b_{3}$, the condition E\bigl{[}\left|\varepsilon_{1}\right|^{2}V_{1}(\varepsilon_{1})\bigr{]}<\infty can be achieved by choosing the value of $b_{1}$ small enough or, specifically, assuming $b_{1}=\tilde{b}_{1}<\beta_{0}$. Furthermore, as $\kappa_{0}\in(0,1]$, by choosing the value of $b_{1}$ small enough the function $k(z_{1})\in(-1,0)$ and is bounded away from $-1$ and [math] for any $\left|z_{1}\right|$ large enough. Therefore, for some $\delta_{1}\in(0,1)$,

[TABLE]

for all sufficiently large $\left|z_{1}\right|$.

Step 2: Bounding $V_{\epsilon}(\boldsymbol{x})E\left[V_{1}\left(g(z_{1})+\varepsilon_{1}\right)\right]$ in (31). For the first term on the right hand side of (31) we obtain, for $\left|z_{1}\right|$ large,

[TABLE]

Write $1-\delta_{1}=(1-\delta_{1})^{1/2}(1-\delta_{1})^{1/2}\leq(1-\tfrac{1}{2}\delta_{1})^{2}$ and note that $1-\tfrac{1}{2}\delta_{1}=\exp\left\{\ln\left(1-\tfrac{1}{2}\delta_{1}\right)\right\}\leq\exp\left\{-\tfrac{1}{2}\delta_{1}\right\}$ to obtain

[TABLE]

As Assumption 1(ii.a) now holds with $d=1$, we have $\left|\epsilon(\boldsymbol{x})\right|=o(\left|\boldsymbol{x}\right|^{-1})$ and

[TABLE]

(cf. the similar inequality in the proof of case $\rho>\kappa_{0}$, Step 2). Therefore, for $\left|z_{1}\right|$ large,

[TABLE]

where

[TABLE]

with $\delta_{1}(z_{1})\in(0,1)$ and $\delta_{1}(z_{1})^{-1}o(\left|\boldsymbol{x}\right|^{-\kappa_{0}})=o(1)$ holding (here, as well as above, the term $o(1)$ is obtained because $\left|z_{1}\right|^{\kappa_{0}}o(\left|\boldsymbol{x}\right|^{-\kappa_{0}})=o(1)$ by arguments similar to those used in the case $\rho>\kappa_{0}$, Step 2).

Thus, we can conclude that, for $\left|z_{1}\right|$ large,

[TABLE]

where the second inequality is due to the convexity of the exponential function. To state this more formally, we can find $M_{1}\geq M_{0}$ and some (finite) $\overline{M}_{1}$, such that

[TABLE]

where $\delta_{1}\in(0,1)$ and $C_{1}=\left\{z_{1}\in\mathbb{R}\,:\,\left|z_{1}\right|\leq M_{1}\right\}$ (cf. the proof of part (ii) in Douc et al. (2004, p. 1373)). Moreover, as in case $\rho>\kappa_{0}$ (the beginning of Step 4), the term $o(1)$ on the right hand side converges to zero as $\left|\boldsymbol{z}_{2}\right|\rightarrow\infty$.

Step 3: Bounding $V_{2}\left(\boldsymbol{\Pi}_{1}\boldsymbol{z}_{2}+z_{1}\boldsymbol{\iota}_{p-1}\right)$ in (31). As in the the proof of case $\rho>\kappa_{0}$, Step 3, assume that the value of $b_{1}$ is fixed to $\tilde{b}_{1}$ specified above. Repeating the arguments in the proof of case $\rho>\kappa_{0}$, Step 3, we first obtain

[TABLE]

where $b_{2,1}=b_{2}\left\|\boldsymbol{\iota}_{p-1}\right\|_{*}^{\kappa_{0}}/\tau_{1}$ and $b_{2,2}=b_{2}/\tau_{2}$ with $\tau_{1}\in(0,1)$ and $\tau_{2}=1-\tau_{1}$. Also, as in case $\rho>\kappa_{0}$, we can choose $\tau_{2}\in(\eta^{\kappa_{0}},1)$ so that $b_{2,2}\eta^{\kappa_{0}}=b_{2}\eta^{\kappa_{0}}/\tau_{2}<b_{2}$, and the value of $b_{2}$ so small that $b_{2,1}=b_{2}\left\|\boldsymbol{\iota}_{p-1}\right\|_{*}^{\kappa_{0}}/\tau_{1}<b_{1}=\tilde{b}_{1}$ with $\tilde{b}_{1}$ as fixed above.

We next bound $V_{2,1}(z_{1})$ and $V_{2,2}(\boldsymbol{z}_{2})$. Arguments similar to those used in the corresponding proof of case $\rho>\kappa_{0}$, Step 3, apply but the bound obtained for $V_{2,1}(z_{1})$ simplifies. Specifically,

[TABLE]

where the first inequality holds for any $\epsilon>0$ as long as $\left|z_{1}\right|$ is large enough and the second inequality holds for some $\eta_{2}\in(0,1)$ and $\left\|\boldsymbol{z}_{2}\right\|_{*}$ bounded away from zero. These inequalities can be written as

[TABLE]

where, for simplicity, we have assumed that the term $\overline{M}_{1}\boldsymbol{1}_{C_{1}}(z_{1})$ can be the same as at the end of Step 2 and where $C_{2}=\left\{\boldsymbol{z}_{2}\in\mathbb{R}^{p-1}\,:\,\left|\boldsymbol{z}_{2}\right|\leq M_{2}\right\}$ with $M_{2}$ and $\overline{M}_{2}$ some positive and finite constants. Thus, we can conclude that

[TABLE]

Step 4: Bounding $E\left[V(\boldsymbol{y}_{1})\,\left|\,\boldsymbol{y}_{0}=\boldsymbol{x}\right.\right]$ in (31). The bounds obtained for $V_{\epsilon}(\boldsymbol{x})E\left[V_{1}\left(g(z_{1})+\varepsilon_{1}\right)\right]$ and V_{2}\bigl{(}\mathbf{\Pi}\boldsymbol{z}_{2}+z_{1}\boldsymbol{\iota}_{p-1}\bigr{)} in Steps 2 and 3, respectively, yield

[TABLE]

As the value of $\epsilon>0$ can be made as close to zero as desired (by only choosing $\left|z_{1}\right|$ large enough and independently of choices made for any other parameters), we can assume that $\epsilon<\tfrac{1}{2}\delta_{1}$ so that

[TABLE]

holds with some $\delta_{2}\in(0,1)$. Moreover, as in the proof of case $\rho>\kappa_{0}$, Step 4,

[TABLE]

$\eta_{3}\in(0,1)$. Thus, defining $\tilde{\lambda}=\delta_{2}\land\eta_{3}\in(0,1)$ and $L\geq 2\overline{M}_{1}+\overline{M}_{2}$ we find that

[TABLE]

We can write the above inequality as

[TABLE]

from which it follows that, for all $\left|\boldsymbol{x}\right|$ large enough, $E\left[V(\boldsymbol{y}_{1})\,\left|\,\boldsymbol{y}_{0}=\boldsymbol{x}\right.\right]\leq(1-\tilde{\lambda})^{\frac{1}{2}}V(\boldsymbol{x})$, implying that there exist positive constants $M$ and $b$ such that, for $C=\left\{\boldsymbol{x}\in\mathbb{R}^{p}\,:\,\left|\boldsymbol{x}\right|\leq M\right\}$,

[TABLE]

Defining $\lambda=1-(1-\tilde{\lambda})^{\frac{1}{2}}\in(0,1)$ we can conclude that Condition D holds with $\phi(v)=\lambda v$ and therefore Theorem 1(i) shows that the Markov chain $\boldsymbol{y}_{t}$ is geometrically ergodic and the convergence (8) holds with $f(\boldsymbol{x})=V(\boldsymbol{x})=V_{1}(z_{1}(\boldsymbol{x}))+V_{2}(\boldsymbol{z}_{2}(\boldsymbol{x}))$.

Case $p=1$: When $p=1$ we have $\boldsymbol{x}=x_{1}=u$ and we simply write $x$ for any of these. In this case, model (16) reduces to $y_{t}=y_{t-1}+\tilde{g}(y_{t-1})+\varepsilon_{t}$, Assumption 1(i) becomes redundant, Assumption 1(ii.a) is automatically satisfied with $g(x)=x+\tilde{g}(x)$, $\epsilon(x)=0$, and $d$ redundant (as long as the condition $\left|g(x)\right|\to\infty$ as $\left|x\right|\to\infty$ still holds), and Assumptions 1(ii.b) and 2 are as in the case $p\geq 2$. In other words, the model can be written as $y_{t}=g(y_{t-1})+\varepsilon_{t}$ with $g$ satisfying Assumption 1(ii.b) as well as $\left|g(x)\right|\to\infty$ as $\left|x\right|\to\infty$. This also means that the assumptions of Theorem 3.3 in Douc et al. (2004) are satisfied except for the continuity of $g$ required in their Assumption 3.4. However, in our case this assumption is not needed because the boundedness of $g$ on compact subsets of $\mathbb{R}$ implied by our Assumption 1(ii) actually suffices.

First consider the case $\rho>\kappa_{0}$. Proceeding as in the proof of Theorem 3.3(i) of Douc et al. (2004) we can conclude that there exist positive constants $M$ and $b$ such that, for $C=\left\{x\in\mathbb{R}\,:\,\left|x\right|\leq M\right\}$ and for all $x\in\mathbb{R}$,

[TABLE]

where $\phi_{1}(V(x))=\tilde{c}_{\phi}\left(1+\ln V(x)\right)^{-\alpha}V(x)$ with $\alpha=\rho/b_{3}-1>0$ and some $\tilde{c}_{\phi}>0$ (see the top of p. 1373 of Douc et al. (2004) and note also our additional assumption $\left|g(x)\right|\to\infty$ as $\left|x\right|\to\infty$). Comparing this with (38) and (39) at the end of the proof of part (i) shows that we can continue as therein and conclude that Condition D is satisfied with $V(x)=\exp\{b_{1}\left|x\right|^{b_{3}}\}$ and $\phi(v)=c(v+v_{0})(\ln(v+v_{0}))^{-\alpha}$ (for some $c,v_{0}>0$ and $\alpha=\rho/b_{3}-1>0$). The result of part (i) now follows from Theorem 1(ii).

Now consider the case $\rho=\kappa_{0}$. As in the proof of Theorem 3.3(ii) of Douc et al. (2004) we can conclude that (40) holds with $\phi_{1}\left(V(x)\right)=\lambda V(x)$ and some $\lambda>0$, and with $M$, $b$, and $C$ redefined (see the middle of p. 1373 of Douc et al. (2004) and note again the above-mentioned additional assumption). The result of part (ii) now follows from Theorem 1(i).∎

Proof of Theorem 3. First note that our Assumption 2(b) implies Assumptions (NSS 1) and (NSS 4) of Fort and Moulines (2003). Also, in the same way as in the proof of Theorem 2 we can show that the Markov chain $\boldsymbol{y}_{t}$ is a $\psi$-irreducible and aperiodic $T$-chain with $\psi$ the Lebesgue measure, and that all compact sets of $\mathcal{B}(\mathbb{R}^{p})$ are petite. This, in turn, implies that Assumption (NSS 2) of Fort and Moulines (2003) holds. These facts together with Assumption 1 are used below to verify Assumption (NSS 3) of Fort and Moulines (2003) which enables us to apply Lemma 3 of that paper.

As $V(\boldsymbol{x})=1+\left|z_{1}\right|^{s_{0}}+s_{1}\left\|\boldsymbol{z}_{2}\right\|_{*}^{\alpha s_{0}}$ we have (cf. the beginning of the proof of Theorem 2)

[TABLE]

In this case it appears convenient to start with bounding the latter term on the right hand side.

Step 1: Bounding $s_{1}\left\|\boldsymbol{\Pi}_{1}\boldsymbol{z}_{2}+z_{1}\boldsymbol{\iota}_{p-1}\right\|_{*}^{\alpha s_{0}}$ in (41). First note that $\alpha=1-\rho/s_{0}\in(0,1)$ because $0<\rho<s_{0}$ is assumed. We consider separately the cases where $\alpha s_{0}\leq 1$ and $\alpha s_{0}>1$, and show that there exist constants $\eta_{0}\in(0,1)$ and $\overline{s}_{1}>0$ such that

[TABLE]

holds for both $\alpha s_{0}\leq 1$ and $\alpha s_{0}>1$. Moreover, the value of $\overline{s}_{1}$ can be chosen as close to zero as desired.

First consider the case $\alpha s_{0}\leq 1$ and assume that $s_{1}<1$. Denoting $\tilde{s}_{1}=s_{1}\left\|\boldsymbol{\iota}_{p-1}\right\|_{*}^{\alpha s_{0}}$ we obtain (cf. the proof of Theorem 2, the beginning of Step 3)

[TABLE]

where $\eta\in(0,1)$ by assumption and $\eta_{1}=1-\eta^{\alpha s_{0}}\in(0,1)$ which shows that inequality (42) holds with $\eta_{0}=\eta_{1}$ and $\overline{s}_{1}=\tilde{s}_{1}$. Also, the value of $\tilde{s}_{1}$ can be made as close to zero as desired by choosing $s_{1}$ small enough.

Now consider the case $\alpha s_{0}>1$. Here $s_{1}<1$ is still assumed and $s_{0}>1$ must hold because $\alpha\in(0,1)$. Write

[TABLE]

where $\eta\in(0,1)$ again holds by assumption. Let $\tau_{1}\in(0,1)$ and $\tau_{2}=1-\tau_{1}$, and denote $s_{1,1}=s_{1}^{1/s_{0}}\left\|\boldsymbol{\iota}_{p-1}\right\|_{*}^{\alpha}/\tau_{1}$ and $s_{1,2}=s_{1}^{1/s_{0}}/\tau_{2}$. Then,

[TABLE]

where the second inequality is justified by the convexity of the function $\left|x\right|\mapsto\left|x\right|^{s_{0}}$ for $s_{0}>1$.

Next, as $\eta^{\alpha}<1$, we can choose $\tau_{2}\in(\eta^{\alpha},1)$ so that $s_{1,2}^{s_{0}}\eta^{\alpha s_{0}}=s_{1}\eta^{\alpha s_{0}}/\tau_{2}^{s_{0}}<s_{1}$. Denoting $\eta_{2}=1-\left(\eta^{\alpha}/\tau_{2}\right)^{s_{0}}$ we have $\eta_{2}\in(0,1)$ and

[TABLE]

and we can conclude that

[TABLE]

Thus, inequality (42) holds with $\eta_{0}=\eta_{2}$ and $\overline{s}_{1}=s_{1,1}^{s_{0}}$. Above we fixed the value of $\tau_{2}$, and hence also the value of $\tau_{1}$, but we are still free to choose the value of $s_{1}$ and make $s_{1,1}^{s_{0}}=s_{1}\left\|\boldsymbol{\iota}_{p-1}\right\|_{*}^{\alpha s_{0}}/\tau_{1}^{s_{0}}<1$ as close to zero as desired by choosing $s_{1}$ small enough. From now on, we assume that $\eta_{0}=\eta_{1}\land\eta_{2}$ and $\overline{s}_{1}=\tilde{s}_{1}\lor s_{1,1}^{s_{0}}$ so that inequality (42) applies irrespective of whether $\alpha s_{0}\leq 1$ or $\alpha s_{0}>1$, and the value of $\overline{s}_{1}$ can be chosen arbitrarily close to zero.

Step 2: Bounding $E\left[\left|\overline{g}(\boldsymbol{x})+\varepsilon_{1}\right|^{s_{0}}\right]$ in (41). Consider the cases $s_{0}<1$ and $s_{0}\geq 1$ separately. When $s_{0}<1$, the definition of the function $\overline{g}$, the triangle inequality, and Assumption 1(ii.a) yield

[TABLE]

When $s_{0}\geq 1$, we can use Minkowski’s inequality and obtain

[TABLE]

The next step is to bound the expectation $E\left[\left|g(z_{1})+\varepsilon_{1}\right|^{s_{0}}\right]$. Assumption 1(ii.b) ensures that the function $g$ satisfies the conditions in Assumption (NSS 3) of Fort and Moulines (2003) which (together with other assumptions of the theorem) implies that we can use Lemma 3 of that paper. Thus, as $\alpha s_{0}=s_{0}-\rho$, inequality (36) in that lemma shows that

[TABLE]

where $\tilde{\epsilon}(z_{1})\rightarrow 0$ as $\left|z_{1}\right|\rightarrow\infty$ and $\lambda>0$ (to see this, note that the cases (i)–(iii) in our Theorem 3 correspond to the cases (i)–(iii) in Lemma 3 of Fort and Moulines (2003) so that the result is obtained with $\lambda=s_{0}r$ in cases (i) and (ii) and with $\lambda=s_{0}r-\frac{1}{2}s_{0}(s_{0}-1)E[\varepsilon_{1}^{2}]$, which is positive by assumption, in case (iii)). Thus, the above inequality implies that, for $\left|z_{1}\right|$ large,

[TABLE]

where $\tilde{\lambda}>0$ and, without loss of generality, we can assume that $\tilde{\lambda}\leq 1$ also holds. Note that this inequality holds for both $s_{0}<1$ and $s_{0}\geq 1$; these two cases will be treated separately below.

Case $s_{0}<1$. First recall from the proof of Theorem 2, Step 2, that

[TABLE]

where $\epsilon_{1}(\boldsymbol{x})=c_{*}\epsilon(\boldsymbol{x})$, $c_{*}>0$. Using (43), (45), and the assumption $s_{0}<1$, we find that, for $\left|z_{1}\right|$ large,

[TABLE]

where the last inequality follows because $\alpha=1-\rho/s_{0}$ so that $\alpha s_{0}=s_{0}-\rho$. As $\left|\epsilon_{1}(\boldsymbol{x})\right|=o(\left|\boldsymbol{x}\right|^{-\rho/s_{0}})$ by assumption, we have, for $\left|z_{1}\right|$ large, $\left|\epsilon_{1}(\boldsymbol{x})\right|^{s_{0}}\left|z_{1}\right|^{\rho}=o(\left|\boldsymbol{x}\right|^{-\rho})\left|z_{1}\right|^{\rho}<\tilde{\lambda}$, and thus

[TABLE]

where $\tilde{\lambda}_{1}\in(0,1)$ and $o(1)\rightarrow 0$ as $\left|\boldsymbol{x}\right|\rightarrow\infty$ (the upper bound of $\tilde{\lambda}_{1}$ follows because $\tilde{\lambda}\leq 1$ was assumed above and the term $o(1)$ is obtained as in the proof of Theorem 2, Step 2).

Case $s_{0}\geq 1$. When $s_{0}\geq 1$, inequalities (44) and (45) imply that, for $\left|z_{1}\right|$ large,

[TABLE]

Here the equality is again due to the definition of $\alpha$ which implies $\alpha s_{0}=s_{0}-\rho$, and the last inequality follows because $(1-u)^{a}\leq 1-au$ holds for all $0\leq u,a\leq 1$. As $\left|\epsilon_{1}(\boldsymbol{x})\right|=o(\left|\boldsymbol{x}\right|^{-\rho})$ by assumption, we have, for $\left|z_{1}\right|$ large enough, $\left|\epsilon_{1}(\boldsymbol{x})\right|\left|z_{1}\right|=\left|\epsilon_{1}(\boldsymbol{x})\right|\left|z_{1}\right|^{\rho}\left|z_{1}\right|^{1-\rho}<\frac{\tilde{\lambda}}{s_{0}}\left|z_{1}\right|^{1-\rho}$, and

[TABLE]

where $\tilde{\lambda}_{2}\in(0,1)$. As the term $1-\tilde{\lambda}_{2}\left|z_{1}\right|^{-\rho}$ in (47) is positive, we can write

[TABLE]

and arguments similar to those in the proof of Theorem 2, Step 2, can be used. Thus, we define $\tau_{1}(z_{1})=1-\tfrac{1}{2}\tilde{\lambda}_{2}\left|z_{1}\right|^{-\rho}$ and $\tau_{2}(z_{1})=1-\tau_{1}(z_{1})$, and express inequality (47) as

[TABLE]

From this we can conclude that, for $\left|z_{1}\right|$ large,

[TABLE]

Here the second inequality is due the convexity of the function $\left|x\right|\mapsto\left|x\right|^{s_{0}}$, $s_{0}\geq 1$, and the last one follows because $s_{0}\geq 1$. By the definition of $\tau_{2}(z_{1})$, $\tau_{2}(z_{1})^{-1}\left|\epsilon_{1}(\boldsymbol{x})\right|=(2/\tilde{\lambda}_{2})\left|z_{1}\right|^{\rho}\left|\epsilon_{1}(\boldsymbol{x})\right|$, so that, for some positive constants $A_{1}$ and $A_{2}$,

[TABLE]

Here the first equation is again due to the definition of $\alpha$ and the last one follows because $\left|\epsilon_{1}(\boldsymbol{x})\right|=o(\left|\boldsymbol{x}\right|^{-\rho})$ by assumption. The second inequality follows because $\left|z_{1}\right|\leq\left|\boldsymbol{z}\right|\leq c\left|\boldsymbol{x}\right|$ and similarly with $\left|z_{1}\right|$ replaced by $\left|\boldsymbol{z}_{2}\right|$ (see footnote 6). Hence, as $\alpha s_{0}=s_{0}-\rho$, we find that, for $\left|z_{1}\right|$ large,

[TABLE]

where $o(1)\rightarrow 0$ as $\left|\boldsymbol{x}\right|\rightarrow\infty$.

To combine the cases $s_{0}<1$ and $s_{0}\geq 1$, set $\tilde{\lambda}_{0}=\tilde{\lambda}_{1}\wedge\tfrac{1}{2}\tilde{\lambda}_{2}\in(0,1)$ and conclude from (46) and (48) that, for $\left|z_{1}\right|$ large,

[TABLE]

where $o(1)\rightarrow 0$ as $\left|\boldsymbol{x}\right|\rightarrow\infty$.

Step 3: Bounding (41). First conclude from inequalities (42) and (49) that, for $\left|z_{1}\right|$ large,

[TABLE]

Furthermore, we noted earlier that the value of $s_{1}$, and hence also the value of $\overline{s}_{1}$, can be chosen as close to zero as desired. Therefore, for $\left|\boldsymbol{z}_{2}\right|$ large enough and for some $\overline{\eta}\in(0,1)$,

[TABLE]

where the replacement of $\left\|\boldsymbol{z}_{2}\right\|_{*}^{\alpha s_{0}}$ with $\left\|\boldsymbol{z}_{2}\right\|_{*}^{\alpha^{2}s_{0}}$ is justified because $\alpha\in(0,1)$ (this replacement is needed below). Also, as we can assume that the value of $\overline{s}_{1}$ is so small that $\tilde{\lambda}_{0}-\overline{s}_{1}>0$, we have $-\tilde{\lambda}_{0}\left|z_{1}\right|^{\alpha s_{0}}+\overline{s}_{1}\left|z_{1}\right|^{\alpha s_{0}}=-\overline{\lambda}\left|z_{1}\right|^{\alpha s_{0}}$ where $\overline{\lambda}\in(0,1)$ (the upper bound follows because $\tilde{\lambda}_{0}<1$, as noted above). Thus, we can conclude that, for $\left|z_{1}\right|$ and $\left|\boldsymbol{z}_{2}\right|$ large,

[TABLE]

Now, let $\left\|\boldsymbol{z}_{2}\right\|_{*}$ be so large that $\overline{\eta}\left\|\boldsymbol{z}_{2}\right\|_{*}^{\alpha^{2}}\geq c>1$. Then,

[TABLE]

where $\overline{\eta}\left(1-1/c\right)\in(0,1]$ and, setting $\overline{c}=\overline{\lambda}\land\left(\overline{\eta}\left(1-1/c\right)\right)$, we have $\overline{c}\in(0,1]$ and

[TABLE]

Next note that

[TABLE]

where the first inequality follows because $\alpha\in(0,1)$ and the second one because $s_{1}<1$ by assumption. This implies that

[TABLE]

By the preceding discussion we can find positive (and finite) constants $M_{i}$ and $\overline{M}_{i}$ $(i=1,2)$ such that

[TABLE]

where $C_{1}=\left\{z_{1}\in\mathbb{R}\,:\,\left|z_{1}\right|\leq M_{1}\right\}$ and $C_{2}=\left\{\boldsymbol{z}_{2}\in\mathbb{R}^{p-1}\,:\,\left|\boldsymbol{z}_{2}\right|\leq M_{2}\right\}$.

Step 4: Completing the proof. Using the definition $V(\boldsymbol{x})=1+\left|z_{1}(\boldsymbol{x})\right|^{s_{0}}+s_{1}\left\|\boldsymbol{z}_{2}(\boldsymbol{x})\right\|_{*}^{\alpha s_{0}}$ and letting $L\geq\overline{M}_{1}+\overline{M}_{2}$, we obtain from (50) that

[TABLE]

where $h(\boldsymbol{x})=\overline{c}V(\boldsymbol{x})^{\alpha-1}$.

As $\alpha\in(0,1)$ and $\overline{c}\in(0,1]$, we have $0<h(\boldsymbol{x})\leq\overline{c}$ and $h(\boldsymbol{x})\rightarrow 0$, as $\left|\boldsymbol{x}\right|\rightarrow\infty$. Comparing the above inequality with inequality (37) (see the proof of Theorem 2 (Part (i), Step 4)) and the properties of the function $h(\boldsymbol{x})$ shows that we can verify Condition D with arguments similar to those in the aforementioned proof. Specifically, we need to show that $L<\tfrac{1}{2}h(\boldsymbol{x})\left(1-h(\boldsymbol{x})\right)^{\frac{1}{2}}V(\boldsymbol{x})$ holds for all $\left|\boldsymbol{x}\right|$ large enough. That this holds is seen by noting that (see the definition of $h(\boldsymbol{x})$ above)

[TABLE]

where $V(\boldsymbol{x})^{\alpha}\rightarrow\infty$ and $V(\boldsymbol{x})^{\alpha-1}\rightarrow 0$, as $\left|\boldsymbol{x}\right|\rightarrow\infty$.

Hence, as in the proof of Theorem 2 (Part (i), Step 4) we can conclude that, there exist positive constants $M$ and $b$ such that, for $C=\left\{\boldsymbol{x}\in\mathbb{R}^{p}\,:\,\left|\boldsymbol{x}\right|\leq M\right\}$,

[TABLE]

where $\phi_{1}\left(v\right)=\tfrac{1}{2}h(\boldsymbol{x})V(\boldsymbol{x})=\tfrac{1}{2}\overline{c}v{}^{\alpha}$. This implies that Condition D holds with $\phi=\phi_{1}$. The result follows from Theorem 1 (note that $\alpha=1-\rho/s_{0}$ so that $1-\alpha=\rho/s_{0}$).

**Case $p=1$: **As in the corresponding proof of Theorem 2, we have $\boldsymbol{x}=x_{1}=u$, so we simply write $x$ for any of these and note the following: Model (16) reduces to $y_{t}=y_{t-1}+\tilde{g}(y_{t-1})+\varepsilon_{t}$, Assumption 1(i) becomes redundant, Assumption 1(ii.a) is automatically satisfied with $g(x)=x+\tilde{g}(x)$, $\epsilon(x)=0$, and $d$ redundant (as long as the condition $\left|g(x)\right|\to\infty$ as $\left|x\right|\to\infty$ still holds), and Assumptions 1(ii.b) and 2 are as when $p\geq 2$. In other words, the model can be written as $y_{t}=g(y_{t-1})+\varepsilon_{t}$ with $g$ satisfying Assumption 1(ii.b) as well as $\left|g(x)\right|\to\infty$ as $\left|x\right|\to\infty$. Note further that now $z_{1}(\boldsymbol{x})$ reduces to $x_{1}$ and we simply write $x$ in place of either of these. Also, due to the choice $g(x)=x+\tilde{g}(x)$ we have $\overline{g}(\boldsymbol{x})=g(x)$.

We go through the changes needed in the proof of Theorem 3 in case $p\geq 2$. Note that the equality $V(\boldsymbol{x})=1+\left|z_{1}\right|^{s_{0}}+s_{1}\left\|\boldsymbol{z}_{2}\right\|_{*}^{\alpha s_{0}}$ in case $p\geq 2$ reduces to $V(x)=1+\left|x\right|^{s_{0}}$ by setting $s_{1}=0$. The beginning of the proof until (41) remains valid with (41) reducing to

[TABLE]

Step 1 can be omitted as the term considered therein equals zero. In Step 2, setting $\epsilon(x)=0$ inequalities (43) and (44) remain valid, and so does (45). The numbered inequalities (46)–(49) all hold but in all of them the last term is set to zero. In Step 3, the first inequality holds with $s_{1}$, $\bar{s}_{1}$, and the $o(1)$ term all set to zero. In the following arguments, set $\bar{\eta}=0$ and $\bar{\lambda}=\tilde{\lambda}_{0}$. Now, some slight changes are needed. Set $\bar{c}=\bar{\lambda}/2\in(0,1)$ and assume $\left|x\right|$ is so large that $\left|x\right|^{\alpha s_{0}}\geq 1/\bar{c}$. This implies that

[TABLE]

similarly to the corresponding derivations in Step 3. Therefore, inequality (50) holds with $s_{1}$ and $\overline{M}_{2}$ set to zero. Step 4 remains valid, so that the stated $(f,r)$-ergodicity result is obtained from Theorem 1 with $f=V^{1-\delta(1-\alpha)}=V^{1-\delta\rho/s_{0}}=(1+\left|x\right|^{s_{0}})^{1-\delta\rho/s_{0}}$ and $\delta\in[1,1/(1-\alpha)]$. Denoting, for brevity, $\gamma=1-\delta\rho/s_{0}\in(0,1]$ note that $1+\left|x\right|^{s_{0}-\delta\rho}=1+(\left|x\right|^{s_{0}})^{\gamma}=\{[1+(\left|x\right|^{s_{0}})^{\gamma}]^{1/\gamma}\}^{\gamma}\leq\{C[1+\left|x\right|^{s_{0}}]\}^{\gamma}$ for some finite positive $C$ (due to Loève’s $c_{r}$-inequality) so that the $(f,r)$-ergodicity with $f(x)=1+\left|x\right|^{s_{0}-\delta\rho}$ follows.∎

Proof of Corollary to Theorem 3. First consider the case $p\geq 2$. We find from the proof of Theorem 3 (the beginning of Step 3) that, for $\left|z_{1}\right|$ large,

[TABLE]

where $\overline{s}_{1}$ is so small that $\tilde{\lambda}_{0}-\overline{s}_{1}>0$ holds and $o(1)\rightarrow 0$ as $\left|\boldsymbol{x}\right|\rightarrow\infty$. Hence, defining $\overline{\eta}\in(0,1)$, $\overline{M}_{1}\boldsymbol{1}_{C_{1}}(z_{1})$, and $\overline{M}_{2}\boldsymbol{1}_{C_{2}}(\boldsymbol{z}_{2})$ as in the proof of Theorem 3 (Step 3), we have

[TABLE]

and setting $c_{1}=(\tilde{\lambda}_{0}-\overline{s}_{1})\land\overline{\eta}$,

[TABLE]

As $V(\boldsymbol{x})=1+\left|z_{1}(\boldsymbol{x})\right|^{s_{0}}+s_{1}\left\|\boldsymbol{z}_{2}(\boldsymbol{x})\right\|_{*}^{\alpha s_{0}}$ and $\alpha s_{0}=s_{0}-\rho$ we can write this, for all $\boldsymbol{x}$, as

[TABLE]

From Theorem 14.3.7 of Meyn and Tweedie (2009) we now find that $\pi\left(\left|z_{1}(\boldsymbol{x})\right|^{s_{0}-\rho}+\left\|\boldsymbol{z}_{2}(\boldsymbol{x})\right\|_{*}^{s_{0}-\rho}\right)<\infty$ and, by the equivalence of vector norms in $\mathbb{R}^{p}$, $\pi\left(\left|z_{1}(\boldsymbol{x})\right|^{s_{0}-\rho}+\left|\boldsymbol{z}_{2}(\boldsymbol{x})\right|^{s_{0}-\rho}\right)<\infty$ also holds. Furthermore, as $\left|z_{1}(\boldsymbol{x})\right|^{s_{0}-\rho}+\left|\boldsymbol{z}_{2}(\boldsymbol{x})\right|^{s_{0}-\rho}\geq c_{2}\left(\left|z_{1}(\boldsymbol{x})\right|+\left|\boldsymbol{z}_{2}(\boldsymbol{x})\right|\right)^{s_{0}-\rho}\geq c_{2}\left|\boldsymbol{z}(\boldsymbol{x})\right|^{s_{0}-\rho}$ and $\left|\boldsymbol{z}(\boldsymbol{x})\right|^{s_{0}-\rho}=\left|\boldsymbol{A}\boldsymbol{x}\right|^{s_{0}-\rho}\geq c_{3}\left|\boldsymbol{x}\right|^{s_{0}-\rho}$ for some $c_{2},\,c_{3}\in(0,\infty)$ (that depend on $s_{0}$ and $\rho$), it follows that $\pi\left(\left|\boldsymbol{x}\right|^{s_{0}-\rho}\right)<\infty$.

In the case $p=1$, the above arguments hold if one sets $\overline{s}_{1}=0$, $c_{1}=\tilde{\lambda}_{0}$, and drops all the terms related to $\boldsymbol{z}_{2}$.∎