Subgeometric Rates of Convergence for Discrete Time Markov Chains under Discrete Time Subordination

TL;DR

This paper investigates how certain subgeometric convergence rates of Markov chains are preserved under discrete time subordination, providing new insights into their invariant measure convergence behavior.

Contribution

It establishes conditions under which subgeometric convergence rates are inherited in discrete time Markov chains via subordination, extending continuous-time results.

Findings

Identifies three typical subgeometric convergence rates.

Provides moment estimates for discrete time subordinators.

Shows inheritance of convergence rates under specific conditions.

Abstract

In this note, we are concerned with the subgeometric rate of convergence of a Markov chain with discrete time parameter to its invariant measure in the $f$-norm. We clarify how three typical subgeometric rates of convergence are inherited under a discrete time version of Bochner's subordination. The crucial point is to establish the corresponding moment estimates for discrete time subordinators under some reasonable conditions on the underlying Bernstein function.

Equations

$$\varphi (x)=a+bx+{\int }_{(0,\infty )}\left(1-{\mathrm{e}}^{-xy}\right)\nu (\mathrm{d}y),x>0,$$

$$\varphi (x)=bx+{\int }_{(0,\infty )}\left(1-{\mathrm{e}}^{-xy}\right)\nu (\mathrm{d}y),x\ge 0,$$

$$c(\varphi ,m)=\frac{1}{m!}{\int }_{(0,\infty )}{y}^m{\mathrm{e}}^{-y}\nu (\mathrm{d}y)+\{\begin{array}{ll}b,& \text{if }m=1,\\ 0,& \text{if }m\ge 2.\end{array}$$

$$\sum _{m=1}^{\infty }c(\varphi ,m)=b+\sum _{m=1}^{\infty }\frac{1}{m!}{\int }_{(0,\infty )}{y}^m{\mathrm{e}}^{-y}\nu (\mathrm{d}y)=b+{\int }_{(0,\infty )}\left(1-{\mathrm{e}}^{-y}\right)\nu (\mathrm{d}y)=\varphi (1)=1,$$

$$\text{mathds}P(T_n=m)=\sum _{m_1+\cdots +m_n=m}\prod _{i=1}^{n}c(m_i,\varphi ).$$

$${\int }_E{P}^n(x,\cdot )\Pi (\mathrm{d}x)=\Pi \text{for all n∈mathdsN.}$$

$$P_{\varphi }^n(x,\mathrm{d}y)=\sum _{m=n}^{\infty }{P}^m(x,\mathrm{d}y)\text{mathds}P(T_n=m).$$

$${\Vert P^n(x,\cdot )-\Pi \Vert }_f\le C(x)r(n),x\in E,n\in \text{mathds}N,$$

$$r(n)={\mathrm{e}}^{-\theta n^{\delta }},r(n)=n^{-\beta },r(n)={\log }^{-\gamma }(2+n),$$

$${\Vert P_{\varphi }^n(x,\cdot )-\Pi \Vert }_f\le C(x)r_{\varphi }(n),x\in E,n\in \text{mathds}N$$

$$\nu (\mathrm{d}y)\ge c{y}^{-1-\alpha }\mathrm{d}y$$

$$r_{\varphi }(n)=\exp \left[-C{n}^{\frac{\delta }{\alpha (1-\delta )+\delta }}\right],$$

$$\underset{x\to \infty }{\mathrm{liminf}}\frac{\varphi (x)}{\log x}>0\text{and}\underset{x\downarrow 0}{\mathrm{limsup}}\frac{\varphi (\lambda x)}{\varphi (x)}>1\text{for some λ>1},$$

$$r_{\varphi }(n)={\left[{\varphi }^{-1}\left(\frac{1}{n}\right)\right]}^{\beta }.$$

$$r_{\varphi }(n)={\log }^{-\gamma }(2+n).$$

$$\underset{x\downarrow 0}{\mathrm{limsup}}\frac{\varphi (\lambda x)}{\varphi (x)}>1\text{for all λ>1}.$$

$$x^{\alpha }=\frac{\alpha }{\Gamma (1-\alpha )}{\int }_0^{\infty }\left(1-{\mathrm{e}}^{-xy}\right)y^{-1-\alpha }\mathrm{d}y,x>0,\alpha \in (0,1),$$

$$\varphi (1)=b+c{\alpha }^{-1}\Gamma (1-\alpha )+{\int }_{(0,\infty )}\left(1-{\mathrm{e}}^{-y}\right)\tilde{\nu }(\mathrm{d}y)=1.$$

$$\text{mathds}E{\mathrm{e}}^{-u{S}_t}={\mathrm{e}}^{-t\varphi (u)},u,t\ge 0.$$

$$\text{mathds}E{\mathrm{e}}^{-\theta S_t^{\delta }}\le \exp \left[-C{t}^{\frac{\delta }{\alpha (1-\delta )+\delta }}\right]\text{for all sufficiently large t>1}.$$

$$\text{mathds}E{S}_t^{-\beta }\le C{\left[{\varphi }^{-1}\left(\frac{1}{t}\right)\right]}^{\beta }\text{for all sufficiently large t>1}.$$

$$\text{mathds}E{\mathrm{e}}^{-\theta T_n^{\delta }}\le \exp \left[-C{n}^{\frac{\delta }{\alpha (1-\delta )+\delta }}\right]\text{for all sufficiently large n∈mathdsN}.$$

$$\text{mathds}E{T}_n^{-\beta }\le C{\left[{\varphi }^{-1}\left(\frac{1}{n}\right)\right]}^{\beta }\text{for all n∈mathdsN}.$$

$$g(x)={\int }_{[0,\infty )}{\mathrm{e}}^{-xt}\mu (\mathrm{d}t)\text{for all x>0}.$$

$$x\mapsto {\mathrm{e}}^{-\theta x^{\delta }},x\mapsto x^{-\beta }$$

$${\mathrm{e}}^{-\theta x^{\delta }}={\int }_0^{\infty }{\mathrm{e}}^{-xt}\psi (\theta ,\delta ,t)\mathrm{d}t,x>0,$$

$$\psi (\theta ,\delta ,t)={\pi }^{-1}{\theta }^{-1/\delta }{\int }_0^{\infty }{\mathrm{e}}^{-{\theta }^{-1/\delta }tu}{\mathrm{e}}^{-u^{\delta }\cos \pi \delta }\sin \left(u^{\delta }\sin \pi \delta \right)\mathrm{d}u;$$

$$x^{-\beta }=\frac{1}{\Gamma (\beta )}{\int }_0^{\infty }{\mathrm{e}}^{-xt}{t}^{\beta -1}\mathrm{d}t,x>0.$$

$$\text{mathds}Eg(T_n)\le \text{mathds}Eg(S_n)\text{for all n∈mathdsN}.$$

$$\text{mathds}Eg(T_n)$$

Full text

Financial support through the National Natural Science Foundation of China (11401442, 11831015) is gratefully acknowledged.

1. Introduction

This article is a continuation of the very recent work [10], where subgeometric rates of convergence are established for continuous time Markov processes under subordination in the sense of Bochner, and it aims to derive the analogous result when the time parameter is discrete. Readers are urged to refer to [7, Chapter 5] and [18, Chapters 13 and 15] for some background on the topic of convergence rates of Markov processes. For recent developments on subgeometric ergodicity, see e.g. [6, 11, 12, 13, 14, 22].

First, we recall the notion of discrete time subordinator, which is a discrete time counterpart of the classical continuous time subordinator (i.e. nondecreasing Lévy process on $[0,\infty)$) and was initialed in [5]; see also [1, 2, 3, 19, 20] for further developments on random walks under discrete time subordination. A function $\phi:(0,\infty)\rightarrow[0,\infty)$ is a Bernstein function if $\phi$ is a $C^{\infty}$-function satisfying $(-1)^{n-1}\phi^{(n)}\geq 0$ for all $n\in\mathds{N}$ (here $\phi^{(n)}$ denotes the $n$-th derivative of $\phi$). It is well known, see e.g. [23, Theorem 3.2], that every Bernstein function has a unique Lévy–Khintchine representation

[TABLE]

where $a\geq 0$ is the killing term, $b\geq 0$ is the drift term and $\nu$ is a Radon measure on $(0,\infty)$ such that $\int_{(0,\infty)}(1\wedge y)\,\nu(\mathrm{d}y)<\infty$. As usual, we make the convention that $\phi(0):=\phi(0+)=a$. For our purpose, we will assume that $\phi$ has no killing term (i.e. $a=0$); that is, $\phi$ is of the form

[TABLE]

where $b$ and $\nu$ are as above. Without loss of generality, we also assume that $\phi(1)=1$; otherwise, we replace $\phi$ by $\phi/\phi(1)$.

For $m\in\mathds{N}$, we set

[TABLE]

Since

[TABLE]

we know that $\{c(\phi,m):m\in\mathds{N}\}$ gives rise to a probability measure on $\mathds{N}$, and hence we can define a random walk $T=\{T_{n}:n\in\mathds{N}\}$ on $\mathds{N}$ by $T_{n}:=\sum_{k=1}^{n}R_{k}$, where $\{R_{k}:k\in\mathds{N}\}$ is a sequence of independent and identically distributed random variables with $\mathds{P}(R_{k}=m)=c(\phi,m)$ for $k,m\in\mathds{N}$. As a strictly increasing process, the random walk $T$ is called a discrete time subordinator associated with the Bernstein function $\phi$. Obviously, $T_{n}\geq n$, and for $m,n\in\mathds{N}$ with $m\geq n$,

[TABLE]

Let $X=\{X_{n}:n\in\mathds{N}\}$ be a Markov chain on a general measurable state space $E$, and denote by $P^{n}(x,\mathrm{d}y)$ the $n$-step transition kernel. Throughout this paper, we always assume that $X$ and $T$ are independent, and that $X$ has an invariant measure $\Pi$:

[TABLE]

The subordinate process is given by the random time-change $X_{n}^{\phi}:=X_{T_{n}}$. The process $X^{\phi}=\{X_{n}^{\phi}:n\in\mathds{N}\}$ is again a Markov chain, and it follows easily from the independence of $X$ and $T$ that the $n$-step transition kernel of $X^{\phi}$ is

[TABLE]

This implies that $\Pi$ is also invariant for the time-changed chain $X^{\phi}$.

For a (measurable) control function $f:E\rightarrow[1,\infty)$, the $f$-norm (cf. [18, Chapter 14]) of a signed measure $\mu$ on $E$ is defined as $\|\mu\|_{f}:=\sup_{|g|\leq f}|\mu(g)|$, where the supremum ranges over all measurable functions $g:E\rightarrow\mathds{R}$ with $|g|\leq f$, and $\mu(g)=\int_{E}g\,\mathrm{d}\mu$. If $f\equiv 1$, then the $f$-norm $\|\cdot\|_{f}$ reduces to the total variation norm $\|\cdot\|_{\operatorname{TV}}$; since $f\geq 1$, we always have $\|\cdot\|_{f}\geq\|\cdot\|_{\operatorname{TV}}$; if furthermore $f$ is bounded then these two norms are equivalent.

It is said that the process $X$ has subgeometric convergence in the $f$-norm if

[TABLE]

where $C(x)$ is a positive constant depending on $x\in E$ and $r:\mathds{N}\rightarrow(0,1]$ is a nonincreasing function with $r(n)\downarrow 0$ and $\log r(n)/n\uparrow 0$ as $n\rightarrow\infty$. Here, $r$ is called the subgeometric rate. In many specific models, the convergence rate $r$ can be explicitly given and typical examples contain

[TABLE]

where $\theta>0$, $\delta\in(0,1)$ and $\beta,\gamma>0$ are some constants. Let $\mathds{Z}_{+}=\mathds{N}\cup\{0\}$ and $\{p_{k}:k\in\mathds{Z}_{+}\}$ be a sequence such that $p_{0}=1$, $p_{k}\in(0,1)$ for all $k\in\mathds{N}$, and $\lim_{k\rightarrow\infty}\prod_{i=1}^{k}p_{i}=0$. Consider the backward recurrence time chain (cf. [18, Section 3.3.1]) on the countable state space $\mathds{Z}_{+}$ with one-step transition kernel $P$ given by $P(k,k+1)=1-P(k,0)=p_{k}$ for all $k\in\mathds{Z}_{+}$. Then this chain admits the convergence rates in (1.4) under some assumptions, see [12, Section 3.1] for details.

In recent years, there has been an increasing interest in the stability of properties of continuous time Markov processes and their semigroups under Bochner’s subordination. See [16] for the dimension-free Harnack inequality for subordinate semigroups, [8] for shift Harnack inequality for subordinate semigroups, [9] for the quasi-invariance property of Brownian motion under random time-change, and [10] for subgeometric rates of convergence for continuous time Markov processes under continuous time subordination. Subordinate functional inequalities can be found in [4, 15, 24].

It is a natural question whether subgeometric rates of convergence can be preserved under discrete time subordination. If $P^{n}$ is subgeometrically convergent to $\Pi$ in the $f$-norm, is it possible to derive quantitative bounds on the convergence rates of the subordinate Markov chain $X^{\phi}$? What we are going to do is to find some function $r_{\phi}:\mathds{N}\rightarrow(0,1]$ such that $\lim_{n\rightarrow\infty}r_{\phi}(n)=0$ and

[TABLE]

for some constant $C(x)>0$ depending only on $x\in E$. As in [10], it turns out that if the convergence rates of the original chain $X$ are of the three typical forms in (1.4), then we are able to obtain convergence rates for the subordinate Markov chain $X^{\phi}$ under some reasonable assumptions on the underlying Bernstein function.

The main result of this note is the following. As usual, denote by $\phi^{-1}$ the inverse function of the (strictly increasing) Bernstein function $\phi$.

Theorem 1.1**.**

Let $X$ be a discrete time Markov chain and $T$ an independent discrete time subordinator associated with Bernstein function $\phi$ given by (1.1) such that $\phi(1)=1$.

1. a)

Assume that (1.3) holds with rate $r(n)=\mathrm{e}^{-\theta n^{\delta}}$ for some constants $\theta>0$ and $\delta\in(0,1]$. If

[TABLE]

for some constants $c>0$ and $\alpha\in(0,1)$, then (1.5) holds with rate

[TABLE]

where $C=C(\theta,\delta,c,\alpha)>0$. 2. b)

Assume that (1.3) holds with rate $r(n)=n^{-\beta}$ for some constant $\beta>0$. If

[TABLE]

then (1.5) holds with rate

[TABLE] 3. c)

Assume that (1.3) holds with rate $r(n)=\log^{-\gamma}(2+n)$ for some constant $\gamma>0$. Then (1.5) holds with rate

[TABLE]

Remark 1.2**.**

a) According to [10, Lemma 2.2 (ii)], the second condition in (1.7) is equivalent to

[TABLE]

b) Let $b=0$ and $\nu(\mathrm{d}y)=\frac{\alpha}{\Gamma(1-\alpha)}\,y^{-1-\alpha}\,\mathrm{d}y$ with $\alpha\in(0,1)$. In this case, it is clear that (1.6) holds, and by the formula [23, p. vii]

[TABLE]

we know that the corresponding Bernstein function (1.1) is given by the fractional power function $\phi(x)=x^{\alpha}$. One can construct more examples for (1.6) by choosing $\nu(\mathrm{d}y)=cy^{-1-\alpha}\,\mathrm{d}y+\tilde{\nu}(\mathrm{d}y)$, where $c>0$, $\alpha\in(0,1)$, and $\tilde{\nu}$ is another Lévy measure on $(0,\infty)$ such that

[TABLE]

c) As pointed out in [10, Remark 1.1], typical examples for Bernstein function $\phi$ satisfying (1.7) are

• •

$\phi(x)=\log(1+x)/\log 2$;

• •

$\phi(x)=x^{\alpha}\log^{\beta}(1+x)/\log^{\beta}2$ with $\alpha\in(0,1)$ and $\beta\in[0,1-\alpha)$;

• •

$\phi(x)=x^{\alpha}\log^{-\beta}(1+x)/\log^{-\beta}2$ with $0<\beta<\alpha<1$;

• •

$\phi(x)=2^{\alpha}x(1+x)^{-\alpha}$ with $\alpha\in(0,1)$.

See [23, Chapter 16] for more examples of such Bernstein functions.

The rest of this paper is organized as follows. Section 2 is devoted to three types of moment estimates for discrete time subordinators, which will be crucial for the proof of Theorem 1.1; we stress that this part is of some interest on its own. In Section 3, we present the proof of Theorem 1.1. Finally, we give in the Appendix an elementary inequality, which has been used in Section 2.

2. Moment estimates for

discrete time subordinators

Recall that a continuous time subordinator $S=\{S_{t}:t\geq 0\}$ associated with Bernstein function $\phi$ is a nondecreasing Lévy process taking values in $[0,\infty)$ and with Laplace transform

[TABLE]

The following result concerning moment estimates for continuous time subordinators is taken from [10, Theorem 2.1].

Lemma 2.1**.**

Let $S$ be a continuous time subordinator associated with Bernstein function $\phi$ given by (1.1).

1. a)

Let $\theta>0$ and $\delta\in(0,1]$. If (1.6) holds for some constants $c>0$ and $\alpha\in(0,1)$, then there exists a constant $C=C(\theta,\delta,c,\alpha)>0$ such that

[TABLE] 2. b)

Let $\beta>0$. If the Bernstein function $\phi$ satisfies (1.7), then there exists a constant $C=C(\beta)>0$ such that

[TABLE]

Analogous to Lemma 2.1, we shall establish the corresponding results for discrete time subordinators. For related moment estimates for general Lévy(-type) processes, we refer to [8, 17].

Our main contribution in this section is the following result.

Theorem 2.2**.**

Let $T$ be a discrete time subordinator associated with Bernstein function $\phi$ given by (1.1) such that $\phi(1)=1$.

1. a)

Let $\theta>0$ and $\delta\in(0,1]$. If (1.6) holds for some constants $c>0$ and $\alpha\in(0,1)$, then there exists a constant $C=C(\theta,\delta,c,\alpha)>0$ such that

[TABLE] 2. b)

Let $\beta>0$. If the Bernstein function $\phi$ satisfies (1.7), then there exists a constant $C=C(\beta)>0$ such that

[TABLE]

In order to prove Theorem 2.2, we first present a general result to bound the completely monotone moment of a discrete time subordinator by that of a continuous time subordinator.

A function $g:(0,\infty)\rightarrow\mathds{R}$ is called a completely monotone function if $g$ is of class $C^{\infty}$ and $(-1)^{n}g^{(n)}\geq 0$ for all $n=0,1,2,\dots$, see [23, Chapter 1]. By the celebrated theorem of Bernstein (cf. [23, Theorem 1.4]), every completely monotone function is the Laplace transform of a unique measure on $[0,\infty)$. More precisely, if $g$ is a completely monotone function, then there exists a unique measure $\mu$ on $[0,\infty)$ such that

[TABLE]

Since the function $x\mapsto x^{\delta}$ ($\delta\in(0,1]$) is a (complete) Bernstein function, it follows easily from [23, Theorem 3.7] that the following functions

[TABLE]

are completely monotone functions, where $\theta>0$, $\delta\in(0,1]$, and $\beta>0$. Indeed, one has for $\theta>0$ and $\delta\in(0,1)$ (see [21]),

[TABLE]

where

[TABLE]

moreover, for $\beta>0$,

[TABLE]

Lemma 2.3**.**

Let $T$ be a discrete time subordinator with Bernstein function $\phi$ given by (1.1) such that $\phi(1)=1$. Let $S$ be a continuous time subordinator with the same Bernstein function $\phi$. If $g:(0,\infty)\rightarrow\mathds{R}$ is a completely monotone function, then

[TABLE]

Proof.

By the representation formula (2.1) and Tonelli’s theorem,

[TABLE]

Note that for $t\geq 0$,

[TABLE]

which does not exceed $\mathrm{e}^{-\phi(t)}$ according to Lemma 4.1 in the Appendix. Then we have for all $n\in\mathds{N}$,

[TABLE]

which was to be proved. ∎

Proof of Theorem 2.2.

Since the functions given in (2.2) are completely monotone functions, we only need to combine Lemma 2.3 with Lemma 2.1 to get the desired estimates. ∎

Since $T_{n}\geq n$, the following lemma is clear.

Lemma 2.4**.**

Let $T$ be a discrete time subordinator associated with Bernstein function $\phi$ given by (1.1) such that $\phi(1)=1$. For any $\gamma>0$ and $n\in\mathds{N}$,

[TABLE]

3. Proof of Theorem 1.1

Lemma 3.1**.**

If (1.3) holds with some rate $r(n)$, then so does (1.5) with rate $r_{\phi}(n)=\mathds{E}\,r(T_{n})$.

Proof.

It holds from (1.2) and (1.3) that

[TABLE]

and hence the claim follows. ∎

Proof of Theorem 1.1.

The assertion follows immediately by combining Lemma 3.1 with the moment estimates for discrete time subordinators derived in Theorem 2.2 and Lemma 2.4. ∎

4. Appendix

If $\phi:[0,\infty)\rightarrow[0,\infty)$ is a concave function, then it is easy to see that

[TABLE]

Lemma 4.1**.**

Let $\phi:[0,\infty)\rightarrow[0,\infty)$ be a concave function such that $\phi(1)=1$ and $\phi$ is differentiable on $(0,1)$ with $\phi^{\prime}|_{(0,1)}\geq 0$. Then

[TABLE]

In particular, the above inequality holds if $\phi$ is a Bernstein function (not necessarily with $\phi(0)=0$).

Proof.

Let

[TABLE]

It follows from (4.1) that for $x\geq 1$,

[TABLE]

whence

[TABLE]

Now we obtain from (4.1) and (4.2) that for $x\geq 1$,

[TABLE]

It remains to consider the case that $x\in[0,1)$. Since $\phi$ is concave and differentiable on $(0,1)$, we know that $\phi^{\prime}$ is nonincreasing on $(0,1)$. For $x\in(0,1)$, by the elementary inequality that $1-\mathrm{e}^{-x}<x$, we obtain

[TABLE]

Moreover, by (4.1) one has for $x\in(0,1)$,

[TABLE]

which yields that

[TABLE]

Combining this with (4.3), we find for $x\in(0,1)$,

[TABLE]

This implies that $\Phi$ is nondecreasing on $(0,1)$ and thus for all $x\in[0,1)$,

[TABLE]

which completes the proof. ∎

Acknowledgements**.**

The author would like to thank an anonymous referee for careful reading and useful suggestions.