Subgeometric ergodicity and $\beta$-mixing

TL;DR

This paper extends the connection between ergodicity and $eta$-mixing from geometric to subgeometric cases, providing new tools for analyzing Markov chains with slower convergence rates.

Contribution

It demonstrates that subgeometric ergodicity implies $eta$-mixing with subgeometric decay, broadening the applicability of mixing results beyond geometric cases.

Findings

Subgeometric ergodicity implies $eta$-mixing with subgeometric decay.

New results for self-exciting threshold autoregressive models.

Useful for analyzing Markov chains where geometric ergodicity fails.

Abstract

It is well known that stationary geometrically ergodic Markov chains are $\beta$-mixing (absolutely regular) with geometrically decaying mixing coefficients. Furthermore, for initial distributions other than the stationary one, geometric ergodicity implies $\beta$-mixing under suitable moment assumptions. In this note we show that similar results hold also for subgeometrically ergodic Markov chains. In particular, for both stationary and other initial distributions, subgeometric ergodicity implies $\beta$-mixing with subgeometrically decaying mixing coefficients. Although this result is simple it should prove very useful in obtaining rates of mixing in situations where geometric ergodicity can not be established. To illustrate our results we derive new subgeometric ergodicity and $\beta$-mixing results for the self-exciting threshold autoregressive model.

Equations

$$\underset{n\to \infty }{\lim }{r}^n\Vert P^n(x;\cdot )-\pi (\cdot )\Vert =0,\text{ πa.e.,}$$

$$\underset{n\to \infty }{\lim }r(n)\Vert P^n(x;\cdot )-\pi (\cdot )\Vert =0,\text{ πa.e..}$$

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

$$\underset{n\to \infty }{\lim }r(n)\Vert P^n(x;\cdot )-\pi (\cdot )\Vert =0,\text{ πa.e..}$$

$$\beta (n)$$

$$\displaystyle=\sup_{m\in\mathbb{N}}\mathsf{E}_{\mu}\Bigl{[}\sup_{B\in\mathcal{F}_{n+m}^{\infty}}\bigl{|}\mathsf{P}_{\mu}(B\mid\mathcal{F}_{0}^{m})-\mathsf{P}_{\mu}(B)\bigr{|}\Bigr{]},$$

$$\beta (n)=\frac{1}{2}\underset{m\in \mathbb{N}}{\sup }{\int }_{\mathsf{X}}\mu P^m(dx)\Vert P^n(x;\cdot )-\mu P^{n+m}(\cdot )\Vert ,n=1,2,\dots ,$$

$$\beta (n)=\frac{1}{2}{\int }_{\mathsf{X}}\pi (dx)\Vert P^n(x;\cdot )-\pi (\cdot )\Vert ,n=1,2,\dots .$$

$$E\left[V(X_1)|X{}_0=x\right]\le V(x)-\beta V(x)+b{1}_C(x),x\in \mathsf{X}.$$

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

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

$$E\left[V(X_1)|X{}_0=x\right]\le V(x)-\varphi (V(x))+b{1}_C(x),x\in \mathsf{X}.$$

$$r_{\varphi }(z)=(H_{\varphi }^{-1}{)}^{\prime }(z)=\varphi \circ H_{\varphi }^{-1}(z).$$

$$X_t=\phi (j)+\theta (j)X_{t-1}+W_t(j),X_{t-1}\in (r_{j-1},r_j],$$

$$\theta (1)<1,\theta (M)<1,\theta (1)\theta (M)<1,$$

$$\theta (1)=1,\theta (M)<1,0<\phi (1),$$

$$\theta (1)<1,\theta (M)=1,\phi (M)<0,$$

$$\theta (1)=1,\theta (M)=1,\phi (M)<0<\phi (1),$$

$$\theta (1)<0,\theta (1)\theta (M)=1,\phi (M)+\phi (1)\theta (M)>0,$$

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

$$\left|g(x)\right|\le \left(1-r/\left|x\right|\right)\left|x\right|\text{for }\left|x\right|\ge M_0\text{and}{\sup }_{|x|\le M_0}\left|g(x)\right|<\infty ;$$

$$\beta (n)\le \frac{1}{2}\int \pi (dx)\Vert P^{n_1}(x;\cdot )-\pi \Vert +\frac{3}{2}\int \mu (dx)\Vert P^{n_1}(x;\cdot )-\pi \Vert ,n=1,2,\dots ,$$

$$\underset{x\in C}{\sup }{\mathsf{E}}_x\left[{\sum }_{k=0}^{{\tau }_C-1}{r}_{\varphi }(k)\right]<\infty ;$$

$$E\left[V_{k+1}(X_1)|X{}_0=x\right]\le V_k(x)-r_{\varphi }(k)+\tilde{b}{r}_{\varphi }(k)1_C(x),$$

$${\mathsf{E}}_x\left[{\sum }_{k=0}^{{\tau }_C-1}{r}_{\varphi }(k)\right]\le V(x)+{\mathsf{E}}_x\left[{\sum }_{k=0}^{{\tau }_C-1}\tilde{b}{r}_{\varphi }(k)1_C(x)\right]=V(x)+\tilde{b}{r}_{\varphi }(0)1_C(x)=V(x)+b\frac{r_{\varphi }(1)}{r_{\varphi }(0)}{1}_C(x).$$

$${\mathsf{E}}_{\mu }\left[r_{\varphi }({\tau }_C)\right]<\infty ,$$

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

$${\tilde{r}}_{\varphi }(n)\beta (n)\le \frac{{\tilde{r}}_{\varphi }(n)}{r_{\varphi }(n_1)}\left[\frac{1}{2}{r}_{\varphi }(n_1)\int \pi (dx)\Vert P^{n_1}(x;\cdot )-\pi \Vert +\frac{3}{2}{r}_{\varphi }(n_1)\int \mu (dx)\Vert P^{n_1}(x;\cdot )-\pi \Vert \right].$$

$$\frac{{\tilde{r}}_{\varphi }(n)}{r_{\varphi }(n_1)}=\frac{{\tilde{r}}_{\varphi }(n)}{{\tilde{r}}_{\varphi ,0}(n)}\frac{{\tilde{r}}_{\varphi ,0}(n)}{r_{\varphi ,0}(n_1)}\frac{r_{\varphi ,0}(n_1)}{r_{\varphi }(n_1)},$$

$$r_{\varphi ,0}(n_1)={\left(\frac{1+\ln (n_1)}{1+\ln (n)}\right)}^{\alpha }(1+\ln (n){)}^{\alpha }\cdot {\left(\frac{1+n_1}{1+n}\right)}^{\beta }(1+n{)}^{\beta }\cdot \frac{e^{c{n}_1^{\gamma }}}{e^{c(n/2{)}^{\gamma }}}{e}^{(c{2}^{-\gamma })n{}^{\gamma }},$$

$$g(x)$$

$$=x+\phi (1)1\{x\le r_1\}+\phi (M)1\{r_{M-1}<x\}+{\sum }_{j=2}^{M-1}[\phi (j)+\theta (j)x-x]1\{x\in (r_{j-1},r_j]\}$$

$${\sup }_{x\in C}{\mathsf{E}}_x\left[r^{{\tau }_C}\right]<\infty ;$$

Full text

Classifications (MSC2010): 60J05, 37A25.

Keywords: Markov chains; rates of convergence, mixing coefficients, subgeometric rate, subexponential rate, polynomial rate, SETAR model.

1 Introduction

Let $X_{t}$ ($t=0,1,2,\ldots$) be a Markov chain on the state space $\mathsf{X}$ with $n$-step transition probability measure $P^{n}$ and stationary distribution $\pi$. If the $n$-step probability measures $P^{n}$ converge in total variation norm to the stationary probability measure $\pi$ at rate $r^{n}$ (for some $r>1$), that is,

[TABLE]

the Markov chain is said to be geometrically ergodic. It is well known that for stationary Markov chains, geometric ergodicity implies that so-called $\beta$-mixing coefficients (or coefficients of absolute regularity) $\beta(n)$, to be defined formally in Section 2, converge to zero at the same rate, $\lim_{n\to\infty}r^{n}\beta(n)=0$ (see, e.g., Doukhan (1994, p. 89), Bradley (2005, Thm 3.7), or Bradley (2007, Thm 21.19)). For initial distributions other than the stationary one, a similar mixing result has been obtained by Liebscher (2005, Propn 4).

We are interested in counterparts of these mixing results when the convergence in (1) takes place at a rate $r(n)$ slower than geometric, that is,

[TABLE]

When (2) holds with suitably defined rates $r(n)$ slower than geometric, the Markov chain is called subgeometrically ergodic. The main result of this note establishes that for both stationary and other initial distributions, subgeometric ergodicity implies $\beta$-mixing with subgeometrically decaying mixing coefficients, that is, $\lim_{n\to\infty}\tilde{r}(n)\beta(n)=0$ for some rate function $\tilde{r}(n)$.

To illustrate some common rate functions, consider the expression

[TABLE]

In the case $\alpha,\beta,c,d>0$ the four terms above satisfy $e^{dn}/e^{cn^{\gamma}}\to\infty$, $e^{cn^{\gamma}}/(1+n)^{\beta}\to\infty$, and $(1+n)^{\beta}/(1+\ln(n))^{\alpha}\to\infty$ as $n\to\infty$, and this hierarchy can be used to define different growth rates. Ordered from the fastest to the slowest growth rate, a growth rate is called geometric (sometimes also exponential) if the dominant term is $e^{dn}$ (with $d>0$; note that $e^{dn}=r^{n}$ with $r>1$ iff $d>0$), subexponential if the dominant term is $e^{cn^{\gamma}}$ ($c>0$ and above $d=0$), polynomial if the dominant term is $(1+n)^{\beta}$ ($\beta>0$, $c=d=0$), and logarithmic if the dominant term is $(1+\ln(n))^{\alpha}$ ($\alpha>0$, $\beta=c=d=0$).

To provide some brief background on subgeometric ergodicity, we note that the first subgeometric ergodicity results for general state space Markov chains were obtained by Nummelin and Tuominen (1983) and Tweedie (1983); the subgeometric rate functions $r(n)$ considered were introduced by Stone and Wainger (1967). Tuominen and Tweedie (1994) gave a set of conditions that imply the convergence in (2) and, in particular, formulated a sequence of so-called drift conditions to establish subgeometric ergodicity. Subsequent work by Fort and Moulines (2000), Jarner and Roberts (2002), Fort and Moulines (2003), and Douc et al. (2004) lead to a formulation of a single drift condition to ensure subgeometric ergodicity, paralleling the use of a Foster-Lyapunov drift condition to establish geometric ergodicity (see, e.g., Meyn and Tweedie (2009, Ch 15)).

The rest of the paper proceeds as follows. Section 2 contains necessary mathematical preliminaries. Section 3 reviews the relation of geometric ergodicity and $\beta$-mixing, while the corresponding results in the subgeometric case are given in Section 4. The general results obtained are exemplified in Section 5 where subgeometric ergodicity and $\beta$-mixing results for the self-exciting threshold autoregressive model are presented. Section 6 concludes, and all proofs are given in an Appendix.

2 Preliminaries

To formalize the discussion in the Introduction, consider $X_{t}$ ($t=0,1,2,\ldots$), a time-homogeneous discrete-time Markov chain on a general measurable state space $(\mathsf{X},\mathcal{B}(\mathsf{X}))$. Comprehensive treatments of the relevant Markov chain theory can be found in Meyn and Tweedie (2009) or Douc et al. (2018). Let $\mu$ be any initial measure on $\mathcal{B}(\mathsf{X})$, and suppose that $X_{0}$ has distribution $\mu$. Denote the transition probabilities with $P(x\,;\,A)$ ($x\in\mathsf{X}$, $A\in\mathcal{B}(\mathsf{X})$) and let $(\Omega,\mathcal{F},\mathsf{P}_{\mu})$ denote the probability space of the Markov process $\{X_{0},X_{1},\ldots\}$. As usual, $\mathsf{P}_{x}$ denotes the probability measure corresponding to a fixed initial value $X_{0}=x$ and $P^{n}(x\,;\,A)=\mathsf{P}_{x}(X_{n}\in A)$ ($x\in\mathsf{X}$, $A\in\mathcal{B}(\mathsf{X})$) signifies the $n$-step transition probability measure.

Next consider the rate of convergence of the $n$-step probability measures $P^{n}$ to the stationary probability measure $\pi$. To this end, for any two probability measures $\lambda_{1}$ and $\lambda_{2}$ on $(\mathsf{X},\mathcal{B}(\mathsf{X}))$, the total variation distance is defined as $\lVert\lambda_{1}-\lambda_{2}\rVert=2\sup_{B\in\mathcal{B}(\mathsf{X})}\lvert\lambda_{1}(B)-\lambda_{2}(B)\rvert=\sup_{\lvert h\rvert\leq 1}\lvert\lambda_{1}(h)-\lambda_{2}(h)\rvert$, where the last supremum runs over all $\mathcal{B}(\mathsf{X})$-measurable functions $h:\mathsf{X}\to\mathbb{R}$ bounded in absolute value by 1 and $\lambda_{i}(h)=\int_{\mathsf{X}}\lambda_{i}(dx)h(x)<\infty$. The $n$-step probability measures $P^{n}$ converge in total variation norm to the stationary probability measure $\pi$ at rate $r(n)$, $n\geq 0$, if

[TABLE]

If (3) holds we say that the Markov chain $X_{t}$ is ergodic with rate $r(n)$; geometric ergodicity obtains when $r(n)=r^{n}$ for some $r>1$.

To define the $\beta$-mixing coefficients, let $\mathcal{F}_{k}^{l}$, $0\leq k\leq l\leq\infty$, signify the $\sigma$-algebra generated by $\{X_{k},\ldots,X_{l}\}$. For the stochastic process $\{X_{0},X_{1},\ldots\}$ the $\beta$-mixing coefficients $\beta(n)$, $n=1,2,\ldots$, are defined as (Doukhan (1994, Sec 1.1); Bradley (2007, Ch 3))

[TABLE]

where $\mathbb{N}=\{0,1,2,\ldots\}$ and in the first expression for $\beta(n)$ the second supremum is taken over all pairs of (finite) partitions $\{A_{1},A_{2},\ldots,A_{I}\}$ and $\{B_{1},B_{2},\ldots,B_{J}\}$ of $\Omega$ such that $A_{i}\in\mathcal{F}_{0}^{m}$ for each $i$ and $B_{j}\in\mathcal{F}_{n+m}^{\infty}$ for each $j$. For our purposes it is convenient to use the following alternative expression obtained by Davydov (1973, Propn 1; note that his definition of $\beta(n)$ includes an additional factor of 2):

[TABLE]

where $\mu P^{m}(\cdot)=\int_{\mathsf{X}}\mu(dx)P^{m}(x\,;\,\cdot)$ denotes the distribution of $X_{m}$ ($m=1,2,\ldots$; $\mu P^{0}=\mu$). In case of a stationary Markov chain (i.e., one with initial distribution $\pi$), the $\beta$-mixing coefficients can be expressed simply as

[TABLE]

Process $X_{t}$ is said to be $\beta$-mixing (or sometimes absolutely regular) if $\lim_{n\rightarrow\infty}\beta(n)=0$. As with the convergence in (3), the rate of this convergence is of interest, and in what follows we seek for results of the form $\lim_{n\rightarrow\infty}r(n)\beta(n)=0$ with some rate function $r(n)$.

3 The geometric case

We start by briefly discussing the relation of geometric ergodicity and $\beta$-mixing; although these results are well known, comparing them with the subgeometric case will be illuminating. In case of a stationary Markov chain (i.e., one with initial distribution $\pi$), this relation is particularly simple. As was first shown by Nummelin and Tuominen (1982, Thm 2.1), a geometrically ergodic Markov chain satisfies, for some $r>1$, $\lim_{n\rightarrow\infty}r^{n}\int\pi(dx)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$; given expression (5), the $\beta$-mixing property immediately follows and the mixing coefficients satisfy $\lim_{n\rightarrow\infty}r^{n}\beta(n)=0$. Statements of this result can be found for instance in Doukhan (1994, p. 89), Bradley (2005, Thm 3.7), and Bradley (2007, Thm 21.19). For initial distributions other than the stationary one, a corresponding result seems to have first appeared in Liebscher (2005, Propn 4).

To facilitate comparison with the subgeometric case, we present the ergodicity and mixing results as consequences of a particular drift criterion; as is discussed in Meyn and Tweedie (2009), this is how geometric ergodicity is often established. We use the following traditional Foster-Lyapunov type geometric drift condition (cf. Meyn and Tweedie (2009, Thm 15.0.1)).111As a technical remark, note that in Condition Drift–G we assume the function $V$ to be everywhere finite (i.e., $V\,:\,\mathsf{X}\rightarrow[1,\infty)$) and such that $\sup_{x\in C}V(x)<\infty$. In contrast, in Meyn and Tweedie (2009, Thm 15.0.1) it is only assumed that $V$ is extended-real-valued (i.e., $V\,:\,\mathsf{X}\rightarrow[1,\infty]$) and finite at some one $x_{0}\in\mathsf{X}$. Our stronger requirements hold in most practical applications and lead to more transparent exposition and proofs. Here $\boldsymbol{1}_{C}(\cdot)$ signifies the indicator function of a set $C$.

Condition** (Drift–G).**

Suppose there exist a petite set $C$, constants $b<\infty$, $\beta>0$, and a measurable function $V\,:\,\mathsf{X}\rightarrow[1,\infty)$ such that $\sup_{x\in C}V(x)<\infty$, satisfying

[TABLE]

For the definition of a ‘petite set’ appearing in this condition, and for the concepts of irreducibility and aperiodicity in the theorem below, we refer the reader to Meyn and Tweedie (2009). Theorem 1 summarizes the relation between geometric ergodicity and $\beta$-mixing.

Theorem 1**.**

Suppose $X_{t}$ is a $\psi$-irreducible and aperiodic Markov chain and that Condition Drift–G holds. Then

• (a)

$X_{t}$* is geometrically ergodic, i.e, for some $r_{1}>1$, $\lim_{n\rightarrow\infty}r_{1}^{n}\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$ for all $x\in\mathsf{X}$.*

Suppose further that the initial state $X_{0}$ has distribution $\mu$ such that $\int_{\mathsf{X}}\mu(dx)V(x)<\infty$. Then

• (b)

for some $r_{2}>1$, $\lim_{n\rightarrow\infty}r_{2}^{n}\int_{\mathsf{X}}\mu(dx)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$,

and

• (c)

$X_{t}$* is $\beta$-mixing and the mixing coefficients satisfy, for some $r_{3}>1$, $\lim_{n\rightarrow\infty}r_{3}^{n}\beta(n)=0$.*

Moreover:

• (d)

In the stationary case ($\mu=\pi$) condition $\int_{\mathsf{X}}\pi(dx)V(x)<\infty$ is not needed, (b) and (c) hold with $r_{2}=r_{3}$, and (b) and (c) are equivalent.

Parts (a) and (b) are very well known (see for instance Meyn and Tweedie (2009, Thm 15.0.1) for part (a) and Nummelin and Tuominen (1982, Thm 2.3) for part (b)) and so is also the mixing result in the stationary case (see the references given earlier). Part (c) for general initial distributions was obtained by Liebscher (2005, Propn 4), although our formulation is somewhat different from his (our formulation and proof avoid the use of so-called ‘$Q$-geometric ergodicity’ employed by Liebscher; for completeness, our proof of Theorem 1, which may be of independent interest, is provided in a Supplementary Appendix). Part (d) elaborates parts (b) and (c) as well as their relation in the stationary case.

4 The subgeometric case

We seek a counterpart of Theorem 1 in which the geometric rate $r^{n}$ is replaced by some slower rate function; such rate functions were already exemplified in the Introduction. More formally, 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 $\alpha>0$.

In analogy with the geometric case, subgeometric ergodicity and mixing results are most conveniently obtained by verifying an appropriate drift condition. The following drift condition for subgeometric ergodicity is adapted from Douc et al. (2018, Defn 16.1.7).222A somewhat more general drift condition, for instance allowing for $V$ to be extended-real-valued, is given in Douc et al. (2004).

Condition** (Drift–SubG).**

Suppose there exist a petite set $C$, a constant $b<\infty$, a concave increasing continuously differentiable function $\phi\,:\,[1,\infty)\rightarrow(0,\infty)$ satisfying $\lim_{v\rightarrow\infty}\phi^{\prime}(v)=0$, and a measurable function $V\,:\,\mathsf{X}\rightarrow[1,\infty)$ such that $\sup_{x\in C}V(x)<\infty$ and

[TABLE]

Note that if $\phi(v)=\eta v$ ($\eta>0$), one obtains Condition Drift–G (but assumption $\lim_{v\rightarrow\infty}\phi^{\prime}(v)=0$ rules this out; as we are interested in subgeometric rates of ergodicity, assuming this means no loss of generality, see Douc et al. (2018, Remark 16.1.8)).

Following Douc et al. (2004) we next introduce a rate function, denoted by $r_{\phi}$. First define the function $H_{\phi}(v)=\int_{1}^{v}\frac{dx}{\phi(x)}$, where $\phi$ is as in Condition Drift–SubG. The definition implies that $H_{\phi}$ is a nondecreasing, concave, and differentiable function on $[1,\infty)$, and it has an inverse $H_{\phi}^{-1}\,:\,[0,\infty)\rightarrow[1,\infty)$ which is increasing and differentiable (see Douc et al. (2004, Sec 2.1)). Thus, we can define the rate function

[TABLE]

Douc et al. (2004, Lemma 2.3 and Proposition 2.5) show that this rate function is subgeometric and that Condition Drift–SubG implies the convergence (3) at rate $r_{\phi}(n)$.

Theorem 2 summarizes the relation between subgeometric ergodicity and $\beta$-mixing. Here $\lfloor k\rfloor$ denotes the integer part of the real number $k$.

Theorem 2**.**

Suppose $X_{t}$ is a $\psi$-irreducible and aperiodic Markov chain and that Condition Drift–SubG holds. Then

• (a)

$X_{t}$* is subgeometrically ergodic with rate $r_{\phi}(n)$, i.e, $\lim_{n\rightarrow\infty}r_{\phi}(n)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$ for all $x\in\mathsf{X}$.*

Suppose further that the initial state $X_{0}$ has distribution $\mu$ such that $\int_{\mathsf{X}}\mu(dx)V(x)<\infty$. Then

• (b)

$\lim_{n\rightarrow\infty}r_{\phi}(n)\int\mu(dx)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$,

and

• (c)

$X_{t}$* is $\beta$-mixing and the mixing coefficients satisfy $\lim_{n\rightarrow\infty}\tilde{r}_{\phi}(n)\beta(n)=0$ for any rate function $\tilde{r}_{\phi}(n)$ such that $\limsup_{n\rightarrow\infty}\tilde{r}_{\phi}(n)/r_{\phi}(n_{1})<\infty$ where $n_{1}=\lfloor n/2\rfloor$.*

Moreover:

• (d)

In the stationary case ($\mu=\pi$) condition $\int_{\mathsf{X}}\pi(dx)V(x)<\infty$ is not needed, (b) and (c) hold with $r_{\phi}(n)=\tilde{r}_{\phi}(n)$, and (b) and (c) (with $r_{\phi}(n)=\tilde{r}_{\phi}(n)$) are equivalent.

• (e)

If $r_{\phi}(n)$ satisfies (6) with $r_{\phi,0}(n)=(1+\ln(n))^{\alpha}\cdot(1+n)^{\beta}\cdot e^{cn^{\gamma}}$ and $\tilde{r}_{\phi}(n)$ satisfies (6) with $\tilde{r}_{\phi,0}(n)=(1+\ln(n))^{\alpha}\cdot(1+n)^{\beta}\cdot e^{\tilde{c}n^{\gamma}}$ for some $0<\tilde{c}<c2^{-\gamma}$, then $\limsup_{n\rightarrow\infty}\tilde{r}_{\phi}(n)/r_{\phi}(n_{1})<\infty$.

Of the results in Theorem 2, part (a) is given in Proposition 2.5 of Douc et al. (2004). Part (b) can be obtained by combining Theorem 4.1 of Tuominen and Tweedie (1994) and Proposition 2.5 of Douc et al. (2004), but in the proof we make use of the work of Nummelin and Tuominen (1983). Part (c) is new and illuminates the relation between subgeometrically ergodic Markov chains and their $\beta$-mixing properties, thereby providing a counterpart of a result obtained by Liebscher (2005, Propn 4) in the case of geometric ergodicity. Part (d) is analogous to its counterpart in Theorem 1 and provides further insight to parts (b) and (c) whereas part (e) makes part (c) more concrete in the case of the most common rate functions. For completeness, we give a detailed proof in the Appendix.

As discussed in Douc et al. (2004, Sec 2.3) and Meitz and Saikkonen (2019, Thm 1), there is a connection between the function $\phi$ and the rate function $r_{\phi}$, which can be used to find out the latter in particular cases. For instance, polynomial rate functions are associated with cases where the function $\phi$ is of the form $\phi(v)=cv^{\alpha}$ with $\alpha\in(0,1)$ and $c\in(0,1]$, and then the rate obtained is $r_{\phi}(n)=n^{\alpha/(1-\alpha)}$ (an alternative form is $r_{\phi}(n)=n^{\kappa-1}$ with $\kappa=1+\alpha/(1-\alpha)$ already given by Jarner and Roberts (2002)). In the subexponential case the function $\phi$ is such that $v/\phi(v)$ goes to infinity slower than polynomially so that a possibility, given in Meitz and Saikkonen (2019, Thm 1), is $\phi(v)=c(v+v_{0})/[\ln(v+v_{0})]^{\alpha}$ for some $\alpha,c,v_{0}>0$. This results in the rate $r_{\phi}(n)=(e^{d})^{n^{1/(1+\alpha)}}$ for some $d>0$ which is faster than polynomial. A logarithmic rate is an example of a rate slower than polynomial. Then the function $\phi$ is of the form $\phi(v)=c[1+\ln(v)]^{\alpha}$ for some $\alpha>0$ and $c\in(0,1]$, and the resulting rate is $r_{\phi}(n)=[\ln(v)]^{\alpha}$ (see Douc et al. (2004, Sec 2.3)).

Theorem 2 (or 1) also provides information about the moments of the stationary distribution of $X_{t}$. Specifically, once part (a) of Theorem 2 (or 1) has been established, one can deduce from Condition Drift–SubG (or Drift–G) and Theorem 14.3.7 of Meyn and Tweedie (2009) that $\int_{\mathsf{X}}\pi(dx)\phi(V(x))<\infty$ (or $\int_{\mathsf{X}}\pi(dx)V(x)<\infty$). This can be very useful when one aims to apply limit theorems developed for $\beta$-mixing processes where moment conditions are typically assumed.

We close this section by noting that Condition Drift–SubG can also be used to obtain more general ergodicity results than provided in Theorem 2. Without going into details we only mention that Theorem 2.8 of Douc et al. (2004) and Theorem 1 of Meitz and Saikkonen (2019) show how a stronger form of ergodicity, called ($f,r$)-ergodicity, can be established.

5 Example

To illustrate our results we consider the self-exciting threshold autoregressive (SETAR) model studied by Chan et al. (1985). These authors analyzed the model

[TABLE]

where $-\infty=r_{0}<\cdots<r_{M}=\infty$ and for each $j=1,\ldots,M$, $\{W_{t}(j)\}$ is an independent and identically distributed mean zero sequence independent of $\{W_{t}(i)\}$, $i\neq j$, and with $W_{t}(j)$ having a density that is positive on the whole real line. They considered the following conditions

[TABLE]

and showed that the SETAR model is ergodic if and only if one of the conditions (8a)–(8e) holds (Chan et al. 1985, Thm 2.1). Moreover, if $E[|W_{t}(j)|]<\infty$ for each $j$, they showed that condition (8a) ensures geometric ergodicity (Chan et al. 1985, Thm 2.3). To our knowledge, in the cases (8b)–(8e) no results regarding the rate of ergodicity have as yet appeared in the literature and our Theorem 4(b) below indicates that geometric ergodicity may not always hold without stronger assumptions.333Meyn and Tweedie (2009, Sec 11.4.3 and Sec B.2) also discuss the (geometric) ergodicity of the SETAR model (7), reproducing the ergodicity result of Chan et al. (1985, Thm 2.1) as their Proposition 11.4.5. On their p. 541, Meyn and Tweedie (2009) also state that (our additions in brackets) “in the interior of the parameter space [the union of (8a)–(8e)] we are able to identify geometric ergodicity in Proposition 11.4.5 … the stronger form [geometric ergodicity] is actually proved in that result” but no formal proof is given for this statement.

We consider rates of ergodicity and $\beta$-mixing in case (8d) when the autoregressive coefficients $\theta(1)$ and $\theta(M)$ equal unity. For intuition, note that due to nonzero intercept terms $\varphi(1)$ and $\varphi(M)$, both the first and the last regimes exhibit nonstationary random walk type behavior with a drift. As the intercept terms satisfy $\varphi(M)<0<\varphi(1)$, the drift is increasing in the first regime and decreasing in the last regime. This feature prevents the process $y_{t}$ from exploding to (plus or minus) infinity, thereby providing intuition why ergodicity can hold true. It is noteworthy that ergodicity is in no way dependent of the behavior of the process in the middle regimes ($2,\ldots,M-1$) which can exhibit stationary, random walk type (with or without drift), or even explosive behavior.

In their results, Chan et al. (1985) allow for regime dependent distributions for the error term $W_{t}(j)$. To obtain our results for the case (8d), we strengthen the assumptions on the error term and, in particular, assume that the error distribution is the same in each regime (this stronger assumption is needed to apply the results mentioned in the proof of Theorem 3 below, and relaxing it appears less than straightforward). To compensate, we obtain results for a model more general than the SETAR model (7) with (8d). Specifically, we formulate our results in terms of the general nonlinear autoregressive model

[TABLE]

where the function $g\,:\,\mathbb{R}\rightarrow\mathbb{R}$ and the error term $\varepsilon_{t}$ satisfy the following conditions:

• (A1)

$g$ is a measurable function with the property $\left|g(x)\right|\rightarrow\infty$ as $\left|x\right|\rightarrow\infty$ and such that there exist positive constants $r$ and $M_{0}$ such that

[TABLE]

• (A2)

$\{\varepsilon_{t},\,t=1,2,\ldots\}$ is a sequence of independent and identically distributed mean zero random variables that is independent of $X_{0}$ and the distribution of $\varepsilon_{1}$ has a (Lebesgue) density that is bounded away from zero on compact subsets of $\mathbb{R}$.

Model (9) with conditions A1 and A2 is a special case of models considered by Fort and Moulines (2003, Sec 2.2), Douc et al. (2004, Sec 3.3), and Meitz and Saikkonen (2019, Secs 3–4). These authors consider much more general models but for clarity of presentation we have simplified the model as much as possible while still being able to obtain results for the SETAR model (7) with (8d) (the first two of the abovementioned papers consider a multivariate version of (9), whereas the third one considers a higher-order generalization of (9); the inequality constraint for the function $g$ in condition A1 is also more general in these papers where it is only required that $\left|g(x)\right|\leq\left(1-r\left|x\right|^{-\rho}\right)\left|x\right|$ for some $0<\rho\leq 2$).

The following Theorem establishes ergodicity and $\beta$-mixing results for model (9) with varying rates of convergence. The proof (in the Appendix) makes use of results in Fort and Moulines (2003), Douc et al. (2004), and Meitz and Saikkonen (2019) to obtain rates of ergodicity, as well as Theorems 1 and 2 above to obtain rates of $\beta$-mixing (only the subgeometric mixing results in parts (b) and (c) are new).

Theorem 3**.**

Consider model (9) with conditions (A1) and (A2).

• (a)

If E\bigl{[}e^{z_{0}\left|\varepsilon_{1}\right|}\bigr{]}<\infty for some $z_{0}>0$, then $X_{t}$ is geometrically ergodic with convergence rate $r(n)=r_{1}^{n}$ for some $r_{1}>1$. Moreover, if the initial state $X_{0}$ has a distribution such that $E[e^{z\left|X_{0}\right|}]<\infty$ for some $z>0$, then $X_{t}$ is also $\beta$-mixing and the mixing coefficients satisfy, for some $r_{3}>1$, $\lim_{n\rightarrow\infty}r_{3}^{n}\beta(n)=0$.

• (b)

If E\bigl{[}e^{z_{0}\left|\varepsilon_{1}\right|^{\kappa_{0}}}\bigr{]}<\infty for some $z_{0}>0$ and $\kappa_{0}\in(0,1)$, then $X_{t}$ is subexponentially ergodic with convergence rate $r(n)=(e^{c})^{n^{\kappa_{0}}}$ (for some $c>0$). Moreover, if the initial state $X_{0}$ has a distribution such that $E[e^{z\left|X_{0}\right|^{\kappa_{0}}}]<\infty$ for some $z>0$, then $X_{t}$ is also $\beta$-mixing and the mixing coefficients satisfy, for some $\tilde{c}>0$, $\lim_{n\rightarrow\infty}(e^{\tilde{c}})^{n^{\kappa_{0}}}\beta(n)=0$.

• (c)

If $E\left[\left|\varepsilon_{1}\right|^{s_{0}}\right]<\infty$ for either $s_{0}=2$ or $s_{0}\geq 4$, then $X_{t}$ polynomially ergodic with convergence rate $r(n)=n^{s_{0}-1}$. Moreover, if the initial state $X_{0}$ has distribution such that $E\left[\left|X_{0}\right|^{s_{0}}\right]<\infty$, then $X_{t}$ is also $\beta$-mixing and the mixing coefficients satisfy $\lim_{n\rightarrow\infty}n^{s_{0}-1}\beta(n)=0$.

Theorem 3 shows that there is a trade-off between rates of ergodicity and $\beta$-mixing and finiteness of moments of the error term. The fastest geometric rate is obtained when E\bigl{[}e^{z_{0}\left|\varepsilon_{1}\right|}\bigr{]}<\infty ($z_{0}>0$) so that $\varepsilon_{1}$ has finite moments of all orders and the slowest polynomial rate is obtained when only $E\left[\varepsilon_{1}^{2}\right]<\infty$. As discussed after Theorem 2, we also have $\int_{\mathsf{X}}\pi(dx)\phi(V(x))<\infty$ so that there is a similar trade-off between these convergence rates and finiteness of moments of the stationary distribution (expressions of $V$ and $\phi$ are available in the proof of Theorem 3).

Above it was mentioned that Fort and Moulines (2003), Douc et al. (2004), and Meitz and Saikkonen (2019) consider (subgeometric) ergodicity of models more general than (9) with conditions (A1) and (A2). Making use of our Theorems 1 and 2, subgeometric rates of $\beta$-mixing can straightforwardly be obtained also for these more general models. We omit the details for brevity.

In a series of papers, Veretennikov and co-authors also considered the model (9) with function $g$ satisfying $\left|g(x)\right|\leq\left(1-r\left|x\right|^{-\rho}\right)\left|x\right|$ for some $1\leq\rho\leq 2$. Using methods very different from ours, they obtained results on subgeometric ergodicity and subgeometric rates for $\beta$-mixing coefficients. The cases $1<\rho<2$ and $\rho=2$ are considered in Veretennikov (2000), Klokov and Veretennikov (2004, 2005), and Klokov (2007) and are shown to lead to subgeometric rates. For the case $\rho=1$ relevant for the SETAR example, these papers refer to Veretennikov (1988, 1991) and Veretennikov and Gulinskii (1990). A result corresponding to our Theorem 3(a) can be found in Veretennikov and Gulinskii (1990, Thm 1) but subgeometric rates, such as those in our Theorem 3(b) and (c), do not seem to be established in the case $\rho=1$.

We now specialize the results above to the SETAR model (7) with (8d). It is easy to see that this model, with the function $g$ in (9) defined as $g(x)=\sum_{j=1}^{M}[\varphi(j)+\theta(j)x]\bm{1}\{x\in(r_{j-1},r_{j}]\}$ (with $\bm{1}\{\cdot\}$ denoting the indicator function), satisfies the condition in A1. Namely, for $x$ large enough and positive we have $\lvert g(x)\rvert=g(x)=x+\varphi(M)=\lvert x\rvert-(-\varphi(M))$ whereas for $x$ small enough and negative we have $\lvert g(x)\rvert=-g(x)=-x-\varphi(1)=\lvert x\rvert-\varphi(1)$, so that the inequality in A1 holds for $M_{0}>\max\{\left|r_{1}\right|,\left|r_{M-1}\right|\}$ and $r=\min\{\varphi(1),-\varphi(M)\}$ (and the supremum condition is obviously satisfied).

Part (a) of the next theorem simply restates the result of Theorem 3 for the SETAR model (7) with (8d), whereas part (b) establishes that geometric ergodicity cannot hold under the weaker moment assumptions of Theorem 3(b) and (c).

Theorem 4**.**

Consider the SETAR model (7) with the parameters satisfying (8d) and the error terms satisfying $W_{t}(j)=\varepsilon_{t}$ ($j=1,\ldots,M$) with $\varepsilon_{t}$ as in (A2).

• (a)

Sufficient conditions for geometric, subexponential, and polynomial ergodicity and $\beta$-mixing of $X_{t}$ are as in parts (a), (b), and (c) of Theorem 3, respectively.

• (b)

If E\bigl{[}e^{z_{0}\left|\varepsilon_{1}\right|}\bigr{]}=\infty for all $z_{0}>0$, then $X_{t}$ is not geometrically ergodic.

Theorem 4(b) shows that for the SETAR model (7) with (8d), the subgeometric rates of Theorem 3(b) and (c) cannot be improved to a geometric rate unless stronger moment assumptions are made regarding the error term. This result is obtained by making use of a necessary condition for geometric ergodicity of certain specific type of Markov chains in Jarner and Tweedie (2003) (using their necessary condition to obtain this result appears possible only in case (8d) out of (8a)–(8e)).

6 Conclusion

In this note we have shown that subgeometrically ergodic Markov chains are $\beta$-mixing with subgeometrically decaying mixing coefficients. Although this result is simple it should prove very useful in obtaining rates of mixing in situations where geometric ergodicity can not be established. An illustration using the popular self-exciting threshold autoregressive model showed how our results can yield new subgeometric rates of mixing.

Appendix

This Appendix contains the proofs of Theorems 2–4; proof of Theorem 1 is provided in the Supplementary Appendix. Proofs of Theorems 1 and 2 make use of the following handy inequality for the $\beta$-mixing coefficients due to Liebscher (2005, Proposition 3). (Note that our Lemma 1 below includes an additional factor of $\frac{1}{2}$ compared to Liebscher’s Proposition 3; cf. our expression for $\beta(n)$ in (4) and his eqn. (27).) Again, $\lfloor k\rfloor$ denotes the integer part of the real number $k$.

Lemma 1**.**

Suppose $X_{t}$ is a Markov chain with stationary distribution $\pi$ and that the initial state $X_{0}$ has distribution $\mu$. Then

[TABLE]

where $n_{1}=\lfloor n/2\rfloor$.

In the proof below, notation $\mathsf{E}_{\mu}\left[\cdot\right]$ is used for the conditional expectation of a $\mathcal{F}_{0}^{\infty}$-measurable random variable conditioned on the initial state $X_{0}$ with distribution $\mu$. When conditioning is on $X_{0}=x$ the notation $\mathsf{E}_{x}\left[\cdot\right]$ is used; these are connected via $\mathsf{E}_{\mu}\left[\cdot\right]=\int_{\mathsf{X}}\mu(dx)\mathsf{E}_{x}\left[\cdot\right]$. We also define the concept of return time to a measurable set $A$ as $\tau_{A}=\inf\left\{n\geq 1:X_{n}\in A\right\}$. For brevity, in the proof we refer to Nummelin and Tuominen (1983) and Douc, Fort, Moulines, and Soulier (2004) as NT83 and DFMS04, respectively.

Proof of Theorem 2*.*

First note that, due to the assumed irreducibility and aperiodicity, the petite set $C$ in Condition Drift–SubG is small (Meyn and Tweedie (2009, Thm 5.5.7)). We first show that

[TABLE]

by Theorem 2.1 of Tuominen and Tweedie (1994) this implies the subgeometric ergodicity in (a) (for related results implying (a), see also NT83, Theorem 2.7(i); Tweedie (1983), Theorem 1(iii); DFMS04 Proposition 2.5). It is shown in DFMS04 (Proposition 2.1 and Lemma 2.3) that Condition Drift–SubG implies the existence of a sequence of drift functions $V_{k}(x)$, $k=0,1,2,\ldots$, such that, for $k\geq 0$,

[TABLE]

where $\tilde{b}=br_{\phi}(1)(r_{\phi}(0))^{-2}$ (see their Proposition 2.1 and top of their page 1358) and $r_{\phi}\in\Lambda$ (see their Lemma 2.3). Applying Proposition 11.3.2 of Meyn and Tweedie (2009) with $Z_{k}=V_{k}(X_{k})$, $f_{k}(x)=r_{\phi}(k)$, $s_{k}(x)=\tilde{b}r_{\phi}(k)\boldsymbol{1}_{C}(x)$, and stopping time $\tau_{C}$ we obtain (DFMS04, Proposition 2.2, also states this conclusion; note also that by their eqn (2.2) we have $V_{0}(x)\leq V(x)$)

[TABLE]

By the condition $\sup_{x\in C}V(x)<\infty$ (in Condition Drift–SubG), we obtain (10). Now, Theorem 2.1 of Tuominen and Tweedie (1994) ensures that $\lim_{n\to\infty}r_{\phi}(n)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$ so that the subgeometric ergodicity in (a) is established (note that as $V_{0}(x)\leq V(x)$ holds with $V(x)$ assumed everywhere finite, the set $S(f,r)$ in Theorem 2.1 of Tuominen and Tweedie (1994) coincides with $\mathsf{X}$ so that the aforementioned convergence holds for all $x\in\mathsf{X}$).

To prove (b), suppose the initial state $X_{0}$ has distribution $\mu$ such that $\int_{\mathsf{X}}\mu(dx)V(x)<\infty$. We will use Theorems 2.7(i,ii) and 2.2 of NT83, but first we obtain a property of the rate function $r_{\phi}(z)$ (which is well-known for members of $\Lambda_{0}$, but not for members of $\Lambda$). Recall that $r_{\phi}(z)=(H_{\phi}^{-1})^{\prime}(z)=\phi\circ H_{\phi}^{-1}(z)$ so that $r_{\phi}^{\prime}(z)/r_{\phi}(z)=\phi^{\prime}\circ H_{\phi}^{-1}(z)$. As $\phi^{\prime}$ is nonincreasing (see Douc et al. (2004, first paragraph of Sec 2.1)) and $H_{\phi}^{-1}$ is increasing, it follows that $r_{\phi}^{\prime}(z)/r_{\phi}(z)=\phi^{\prime}\circ H_{\phi}^{-1}(z)$ is nonincreasing. Therefore also the function $\ln(r_{\phi}(x))/x=\frac{1}{x}\int_{0}^{x}(r_{\phi}^{\prime}(s)/r_{\phi}(s))ds$ ($x>0$) is nonincreasing. Following the proof of Lemma 1 in Stone and Wainger (1967) (which relies only on their property (iii) on their p. 326) yields the desired property $r_{\phi}(m+n)\leq r_{\phi}(m)r_{\phi}(n)$ for all $m,n>0$.

Using this property we now obtain $r_{\phi}(\tau_{C})\leq r_{\phi}(1)r_{\phi}(\tau_{C}-1)\leq r_{\phi}(1)\sum_{k=0}^{\tau_{C}-1}r_{\phi}(k)$ and further \mathsf{E}_{x}\bigl{[}\sum_{k=0}^{\tau_{C}}r_{\phi}(k)\bigr{]}\leq(r_{\phi}(1)+1)\mathsf{E}_{x}\bigl{[}\sum_{k=0}^{\tau_{C}-1}r_{\phi}(k)\bigr{]} and \mathsf{E}_{x}\left[r_{\phi}(\tau_{C})\right]\leq r_{\phi}(1)\mathsf{E}_{x}\bigl{[}\sum_{k=0}^{\tau_{C}-1}r_{\phi}(k)\bigr{]} (cf. Tuominen and Tweedie (1994, eqns (5) and (14)). The former result together with (10) implies that condition (2.12) of Theorem 2.7(i) of NT83 is satisfied. The latter result together with (11) yields $\mathsf{E}_{x}\left[r_{\phi}(\tau_{C})\right]\leq r_{\phi}(1)[V(x)+b\frac{r_{\phi}(1)}{r_{\phi}(0)}\boldsymbol{1}_{C}(x)]$ and, as $\mathsf{E}_{\mu}\left[r_{\phi}(\tau_{C})\right]=\int_{\mathsf{X}}\mu(dx)\mathsf{E}_{x}\left[r_{\phi}(\tau_{C})\right]$, the assumed bound $\int_{\mathsf{X}}\mu(dx)V(x)<\infty$ implies

[TABLE]

so that the condition in Theorem 2.7(ii) of NT83 is satisfied. Therefore, by Theorems 2.7(i,ii) and 2.2 of NT83,

[TABLE]

Next consider part (d). In the stationary case ($\mu=\pi$) the result $\lim_{n\rightarrow\infty}r_{\phi}(n)\int\pi(dx)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$ follows from the last remark in Theorem 2.2 of NT83 (and condition $\int_{\mathsf{X}}\pi(dx)V(x)<\infty$ is not needed). Thus (b) holds in the stationary case. Regarding part (c) in the stationary case, note from (5) that now $\beta(n)=\int\pi(dx)\left\|P^{n}(x\,;\,\cdot)-\pi\right\|$, $n=1,2,\ldots$, so that (b) and (c) are clearly equivalent (and hold with the same rate $r_{\phi}(n)$).

To prove (c), use Lemma 1 to obtain the inequality

[TABLE]

The term in square brackets converges to zero as $n\to\infty$ by parts (b) and (d) and, by assumption, $\limsup_{n\rightarrow\infty}\tilde{r}_{\phi}(n)/r_{\phi}(n_{1})<\infty$. This establishes (c).

To prove (e), it suffices to note that

[TABLE]

where the first and the last ratio on the right hand side are bounded from above uniformly in $n$ due to (6), and that

[TABLE]

where the three ratios on the right hand side are clearly bounded from below uniformly in $n$ by some constant larger than zero. ∎

Proof of Theorem 3*.*

The ergodicity results of parts (a) and (b) could be obtained using results in Douc et al. (2004, Sec 3.3) and those in part (c) using results in Fort and Moulines (2003, Sec 2.2); for clarity of presentation, we will in all parts rely on the results in Meitz and Saikkonen (2019), henceworth MS19. Model (9) with conditions (A1) and (A2) is a special case of the model considered in MS19 (with $p=\rho=1$ in that paper). Of the assumptions made in MS19, Assumption 1 holds due to A1 and either Assumption 2(a) or 2(b) holds due to A2 and the moment conditions assumed in parts (a)–(c) of Theorem 3. Therefore we can make use Theorems 2 and 3 in MS19 to obtain suitable ergodicity results.

(a) In this case Assumption 2(a) of MS19 holds with $\kappa_{0}=1$ and we apply their Theorem 2(ii). From the proof of that theorem (Case $p=1$) it can be seen that Condition Drift–G holds with $V(x)=e^{b_{1}\left|x\right|}$ for some $b_{1}\in(0,z_{0})$ which can be chosen as small as desired. From Theorem 2(ii) of MS19 we obtain that $X_{t}$ is geometrically ergodic with convergence rate $r(n)=(e^{c})^{n}$ (for some $c>0$), that is, $r(n)=r_{1}^{n}$ for some $r_{1}>1$. To obtain results on $\beta$-mixing, we next apply Theorem 1 of the present paper. If the initial state $X_{0}$ has distribution such that $E[e^{z\left|X_{0}\right|}]<\infty$ for some $z>0$ (and noting that above $b_{1}$ can be chosen small enough so that $b_{1}\leq z$ holds), then by Theorem 1 $X_{t}$ is $\beta$-mixing and the mixing coefficients satisfy, for some $r_{3}>1$, $\lim_{n\rightarrow\infty}r_{3}^{n}\beta(n)=0$.

(b) In this case Assumption 2(a) of MS19 holds with $\kappa_{0}\in(0,1)$ and we apply their Theorem 2(i). From the proof of that theorem (Case $p=1$) it can be seen that Condition Drift–SubG holds with $V(x)=e^{b_{1}\left|x\right|^{\kappa_{0}}}$ (for some $b_{1}\in(0,\beta_{0})$ which can be chosen as small as desired) and $\phi(v)=c_{0}(v+v_{0})(\ln(v+v_{0}))^{-\alpha}$ (for some $c_{0},v_{0}>0$ and $\alpha=1/\kappa_{0}-1$). From Theorem 2(i) of MS19 we obtain that $X_{t}$ is subexponentially ergodic with convergence rate $r(n)=(e^{c})^{n^{\kappa_{0}}}$ (for some $c>0$). To obtain results on $\beta$-mixing, we next apply Theorem 2 of the present paper. If the initial state $X_{0}$ has distribution such that $E[e^{z\left|X_{0}\right|^{\kappa_{0}}}]<\infty$ for some $z>0$ (and noting that above $b_{1}$ can be chosen small enough so that $b_{1}\leq z$ holds), then by Theorem 2 $X_{t}$ is $\beta$-mixing and the mixing coefficients satisfy, for any $\tilde{c}\in(0,z2^{-\kappa_{0}})$, $\lim_{n\rightarrow\infty}\tilde{r}(n)\beta(n)=0$ with $\tilde{r}(n)=(e^{\tilde{c}})^{n^{\kappa_{0}}}$.

(c) In this case Assumption 2(b) of MS19 holds with either $s_{0}=2$ or $s_{0}\geq 4$ and we apply their Theorem 3(ii) (in which exactly the cases $s_{0}=2$ and $s_{0}\geq 4$ are available). From the proof of that theorem (the end of Step 4 and Case $p=1$) it can be seen that Condition Drift–SubG holds with $V(x)=1+\left|x\right|^{s_{0}}$ and $\phi(v)=cv^{\alpha}$ (for some $c>0$ and $\alpha=1-1/s_{0}$). From Theorem 3(ii) of MS19 we obtain that $X_{t}$ is polynomially ergodic with convergence rate $r(n)=n$ ($s_{0}=2$) or $r(n)=n^{s_{0}-1}$ ($s_{0}\geq 4$). To obtain results on $\beta$-mixing, we next apply Theorem 2 of the present paper. If the initial state $X_{0}$ has distribution such that $E\left[\left|X_{0}\right|^{s_{0}}\right]<\infty$, then $X_{t}$ is $\beta$-mixing and the mixing coefficients satisfy $\lim_{n\rightarrow\infty}n\beta(n)=0$ ($s_{0}=2$) or $\lim_{n\rightarrow\infty}n^{s_{0}-1}\beta(n)=0$ ($s_{0}\geq 4$). ∎

Proof of Theorem 4*.*

Part (a) follows immediately from Theorem 3 and the discussion preceding it noting that the SETAR model (7) with (8d) satisfies the condition in A1. To prove (b), assume that E\bigl{[}e^{z_{0}\left|\varepsilon_{1}\right|}\bigr{]}=\infty for all $z_{0}>0$ but that $X_{t}$ would be geometrically ergodic. We will use results of Jarner and Tweedie (2003) to show that this leads to a contradiction. To this end, note that for the SETAR model (7) with the parameters satisfying (8d) the function $g$ in our equation (9) equals $g(x)=\sum_{j=1}^{M}[\varphi(j)+\theta(j)x]\bm{1}\{x\in(r_{j-1},r_{j}]\}$ which can be written as

[TABLE]

or as $g(x)=x+\tilde{g}(x)$ where $\tilde{g}(x)$ is bounded. Also recall that it is assumed that the error terms satisfy $W_{t}(j)=\varepsilon_{t}$ ($j=1,\ldots,M$) with $\varepsilon_{t}$ as in (A2). These facts show that the SETAR model (7) with (8d) can be expressed in the form of equation (3) in Jarner and Tweedie (2003) so that $X_{t}$ is what Jarner and Tweedie (2003) call a “random-walk-type Markov chain”. (Note also that this holds only in case (8d) out of (8a)–(8e).) Theorem 2.2 of Jarner and Tweedie (2003) shows that a necessary condition for the geometric ergodicity of a random-walk-type Markov chain $X_{t}$ with stationary probability measure $\pi$ is that there exists a $z>0$ such that $\int_{\mathsf{\mathbb{R}}}e^{z\left|x\right|}\pi(dx)<\infty$. This can be shown to be in contradiction with our assumption that E\bigl{[}e^{z_{0}\left|\varepsilon_{1}\right|}\bigr{]}=\infty for all $z_{0}>0$ as follows.

Suppose $z>0$ is such that $\int_{\mathsf{\mathbb{R}}}e^{z\left|x\right|}\pi(dx)<\infty$ and assume that $X_{0}$, and hence also $X_{1}$, has the stationary distribution $\pi$. Thus $E[e^{z\left|X_{0}\right|}]<\infty$ and $E[e^{z\left|X_{1}\right|}]<\infty$. As $0<e^{zx}\leq e^{z\left|x\right|}$ and $0<e^{-zx}\leq e^{z\left|x\right|}$, it follows that $E[e^{zX_{0}}]$, $E[e^{-zX_{0}}]$, $E[e^{zX_{1}}]$, and $E[e^{-zX_{1}}]$ are all positive and finite. As $X_{1}=X_{0}+\tilde{g}(X_{0})+\varepsilon_{1}$ with $X_{0}$ and $\varepsilon_{1}$ independent, $E[e^{zX_{1}}]=E[e^{zX_{0}}e^{z\tilde{g}(X_{0})}]E[e^{z\varepsilon_{1}}]$ (due to the nonnegativity of the exponential function, this holds whether the expectations involved are finite or equal $+\infty$). As $0<E[e^{zX_{0}}],E[e^{zX_{1}}]<\infty$ and $\tilde{g}(X_{0})$ is bounded this implies that $0<E[e^{z\varepsilon_{1}}]<\infty$. An analogous argument yields that $0<E[e^{-z\varepsilon_{1}}]<\infty$. Finally, nonnegativity of the random variables involved implies that $E[e^{z\left|\varepsilon_{1}\right|}]=E[e^{z\varepsilon_{1}}\bm{1}\{\varepsilon_{1}\geq 0\}+e^{-z\varepsilon_{1}}\bm{1}\{\varepsilon_{1}<0\}]\leq E[e^{z\varepsilon_{1}}]+E[e^{-z\varepsilon_{1}}]<\infty$, yielding a contradiction. ∎

Supplementary Appendix     (not meant for publication)

Proof of Theorem 1*.*

For brevity, we refer to Nummelin and Tuominen (1982) and Meyn and Tweedie (2009) as NT82 and MT09, respectively. We use Theorem 2.5(ii) of NT82 to prove (a). To this end, first note that, due to the assumed irreducibility and aperiodicity, the petite set $C$ in Condition Drift–G is small (MT09, Theorem 5.5.7). We first show that, for some $r>1$,

[TABLE]

cf. Theorem 2.5(ii) of NT82. We proceed as in the proof of Theorem 15.2.5 in MT09 and, for the $\beta$ in Condition Drift–G, choose an $r\in(1,(1-\beta)^{-1})$ and set $\varepsilon=r^{-1}-(1-\beta)$ so that $0<\varepsilon<\beta$ and $\varepsilon$ is the solution to $r=(1-\beta+\varepsilon)^{-1}$. Now we may reorganize the drift condition as

[TABLE]

Define $Z_{k}=r^{k}V(X_{k})$, $k=0,1,2,\ldots$, so that $E[Z_{k+1}\mid\mathcal{F}_{0}^{k}]=r^{k+1}E[V(X_{k+1})\mid\mathcal{F}_{0}^{k}]$ and thus

[TABLE]

Applying Proposition 11.3.2 of MT09 with $f_{k}(x)=\varepsilon r^{k+1}V(x)$, $s_{k}(x)=br^{k+1}\boldsymbol{1}_{C}(x)$, and stopping time $\tau_{C}$ we obtain

[TABLE]

because $\boldsymbol{1}_{C}(X_{1})=\cdots=\boldsymbol{1}_{C}(X_{\tau_{C}-1})=0$ by the definition of $\tau_{C}$. Multiplying by $\varepsilon^{-1}r^{-1}$ and noting that $V(\cdot)\geq 1$, we obtain, for some finite constants $c_{1},c_{2}$,

[TABLE]

As $\sup_{x\in C}V(x)<\infty$, \sup_{x\in C}\mathsf{E}_{x}\bigl{[}\sum_{k=0}^{\tau_{C}-1}r^{k}\bigr{]}<\infty is obtained. Using $\sum_{k=0}^{\tau_{C}-1}r^{k}=(r^{\tau_{C}}-1)/(r-1)$, this is equivalent to $\sup_{x\in C}\mathsf{E}_{x}\left[r^{\tau_{C}}\right]<\infty$ as desired; note that we also have, for some finite constants $c_{3},c_{4}$, $\mathsf{E}_{x}\left[r^{\tau_{C}}\right]\leq c_{3}V(x)+c_{4}$. Now Theorem 2.5(ii) of NT82 implies that, for some $r_{1}>1$, $\lim_{n\rightarrow\infty}r_{1}^{n}\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$, so that the geometric ergodicity of part (a) is established.

To prove (b), suppose the initial state $X_{0}$ has distribution $\mu$ such that $\int_{\mathsf{X}}\mu(dx)V(x)<\infty$. By Theorem 2.5(iii) of NT82 it suffices to prove that $\mathsf{E}_{\mu}\left[r^{\tau_{C}}\right]<\infty$. As $\mathsf{E}_{\mu}\left[r^{\tau_{C}}\right]=\int_{\mathsf{X}}\mu(dx)\mathsf{E}_{x}\left[r^{\tau_{C}}\right]$, the inequality $\mathsf{E}_{x}\left[r^{\tau_{C}}\right]\leq c_{3}V(x)+c_{4}$ obtained above implies $\mathsf{E}_{\mu}\left[r^{\tau_{C}}\right]<\infty$ and hence the validity of (b) for some $r_{2}>1$ (Theorem 2.5(iii) of NT82).

Next consider part (d). In the stationary case ($\mu=\pi$), the geometric ergodicity established in (a) and Theorem 2.1 of NT82 imply that $\lim_{n\rightarrow\infty}\tilde{r}_{2}^{n}\int\pi(dx)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|=0$ for some $\tilde{r}_{2}>1$ (and condition $\int_{\mathsf{X}}\pi(dx)V(x)<\infty$ is not needed). Thus (b) holds in the stationary case. Regarding part (c) in the stationary case, note from (5) that now $\beta(n)=\int\pi(dx)\left\|P^{n}(x\,;\,\cdot)-\pi(\cdot)\right\|$, $n=1,2,\ldots$, so that (b) and (c) are clearly equivalent (and hold with the same rate $\tilde{r}_{2}$).

To prove (c) in the general case, recall that $n_{1}=\lfloor n/2\rfloor$ so that $n/2-1<n_{1}\leq n/2,$ and note that for any $\rho>1$ and $n\geq 2$, $1=\rho^{1-n/2}\rho^{n/2-1}<\rho^{1-n/2}\rho^{n_{1}}=\rho(\rho^{1/2})^{-n}\rho^{n_{1}}$. Now choose $r_{3}$ such that $1<r_{3}<\min\{r_{2}^{1/2},\tilde{r}_{2}^{1/2}\}$ (where $r_{2}$ and $\tilde{r}_{2}$ are as above in the proofs of parts (b) and (d)). Now use these remarks and the inequality in Lemma 1 to obtain

[TABLE]

From the proofs of (b) and (d) we obtain the results $\lim_{n\rightarrow\infty}r_{2}^{n_{1}}\int\mu(dx)\left\|P^{n_{1}}(x\,;\,\cdot)-\pi\right\|=0$ and $\lim_{n\rightarrow\infty}\tilde{r}_{2}^{n_{1}}\int\pi(dx)\left\|P^{n_{1}}(x\,;\,\cdot)-\pi\right\|=0$, so that $\lim_{n\rightarrow\infty}r_{3}^{n}\beta(n)=0$ and hence (c) follows. ∎