Convergence Rate of Euler–Maruyama Scheme to the Invariant Probability Measure Under Total Variation Distance for the SDEs

Abstract

This article shows the geometric decay rate of the Euler–Maruyama scheme for a one-dimensional stochastic differential equation towards its invariant probability measure under total variation distance. Firstly, the existence and uniqueness of invariant probability measure and the uniform geometric ergodicity of the chain are studied through the introduction of non-atomic Markov chains. Secondly, the equivalent conditions for uniform geometric ergodicity of the chain are discovered by constructing a split Markov chain based on the original Euler–Maruyama scheme.

1. Introduction

Consider the following stochastic differential equation (SDE) on $\mathbb{R}$:

$$d{X}_t=g\left(X_t\right)dt+\sigma d{B}_t,X_0=x_0,$$

(1)

where ${\left(B_t\right)}_{t⩾0}$ is a standard Brownian motion, $\sigma >0$ is a constant, and $g:\mathbb{R}\to \mathbb{R}$ satisfies the hypothesis below.

Hypothesis (H1).

There exist L, $K_1>0$ and $K_2⩾0$ such that for every $(x,y)\in {\mathbb{R}}^2$,

$$\begin{array}{cc}\hfill \left|g\right(x)-g(y\left)\right|& ⩽L|x-y|,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill \left(g\right(x)-g(y\left)\right)(x-y)& ⩽-K_1{(x-y)}^2+K_2.\hfill \end{array}$$

(3)

Moreover, g is second-order differentiable, and the second-order derivative of g is bounded.

The inequality (2) implies for any $x\in \mathbb{R}$,

$$g^2\left(x\right)⩽2L^2{x}^2+2g^2\left(0\right),\left|g^{\prime }\left(x\right)\right|⩽L,$$

(4)

where $g^{\prime }$ represents the derivative of the function g. Moreover, the inequality (3) and Young’s inequality imply

$$xg\left(x\right)⩽-{\displaystyle \frac{K_1}{2}}{x}^2+c_g,$$

(5)

with $c_g=K_2+{\displaystyle \frac{1}{2K_1}}{g}^2\left(0\right)$.

For a given step size $\eta \in (0,1)$, the Euler–Maruyama (EM) scheme ${\left({\theta }_k\right)}_{k⩾0}$ for the SDE (1) is given by the following recursive relation: for any $k⩾0$,

$${\theta }_{k+1}={\theta }_k+\eta g\left({\theta }_k\right)+\sqrt{\eta }\sigma {\epsilon }_{k+1},$$

(6)

where ${\left({\epsilon }_k\right)}_{k⩾1}$ are independent and identically distributed standard normal random variables.

Over the past decades, EM schemes have been widely and intensively used for solution approximations of SDEs on ${\mathbb{R}}^d$ with $d⩾1$ (see for instance [1,2,3,4,5,6,7,8]). In the case when $g\left(x\right)=-\nabla U\left(x\right)$ with U being a potential, (1) is called Langevin diffusion, and (6) is called unadjusted Langevin algorithm (ULA) with constant step size. The ULA is nowadays a popular class of Markov chain Monte Carlo (MCMC) algorithms ([9]), which is commonly applied to solve Monte Carlo sampling problems especially in the field of machine learning (see, for instance, [10,11]). Roberts and Tweedie [12] initially provided probabilistic analysis of asymptotic behaviour of Langevin diffusion and ULA. They found necessary (Theorem 2.2, [12]) and sufficient conditions (Theorem 2.3, [12]) for exponential ergodicity of Langevin diffusion, and sufficient conditions for geometric ergodicity of ULA (Theorem 3.1, [12]) respectively. Dalalyan [13] found an upper bound for the error in total variation (TV) distance between ${\theta }_k$ and the ergodic measure of ${\left(X_t\right)}_{t⩾0}$. Durmus and Moulines [14] established geometric ergodicity in TV distance for Langevin diffusion under similar convexity and Lipschitz conditions, with explicit convergence rates. Recently, Fan et al. [15] established normalized and self-normalized Cramér-type moderate deviations for the EM scheme for the SDE (1) on ${\mathbb{R}}^d$ with $d⩾1$. Meanwhile, the geometric ergodicity of the ULA towards its invariant probability measure under appropriate distance has been rarely studied so far in the literature. As an extension of classical Markov chains with discrete state space, the EM scheme studied in this paper is a non-atomic Markov chain, which is also of independent interest for the study on other stochastic models with a non-independence structure.

The objectives of this work are to prove the geometric ergodicity of the EM scheme given by (6) for the SDE (1) under TV distance, and provide furthermore a convergence rate to its invariant probability measure.

Let $\mathcal{B}\left(\mathbb{R}\right)$ be the Borel $\sigma$-algebra on $\mathbb{R}$. From (6), it can be seen that ${\left({\theta }_k\right)}_{k⩾0}$ is a Markov chain on the continuous state space $\mathbb{R}$, with Markov kernel $P_{\eta }$ given by

$$\begin{array}{c}\hfill P_{\eta }(x,A)={\displaystyle \frac{1}{\sqrt{2\pi \eta }\sigma }}{\int }_{A}exp\left(-{\displaystyle \frac{{(y-x-\eta g\left(x\right))}^2}{2\eta {\sigma }^2}}\right)dy,\end{array}$$

(7)

for any $x\in \mathbb{R}$ and $A\in \mathcal{B}\left(\mathbb{R}\right)$. From (7), $P_{\eta }$ admits a kernel density with respect to (w.r.t) the Lebesgue measure given by $p_{\eta }(x,y):={\displaystyle \frac{1}{\sqrt{2\pi \eta }\sigma }}exp\left(-{\displaystyle \frac{{(y-x-\eta g\left(x\right))}^2}{2\eta {\sigma }^2}}\right)$. The Markov kernel $P_{\eta }$ can be defined equivalently as, for any $x\in \mathbb{R}$ and measurable function h on $\left(\mathbb{R},\mathcal{B}\left(\mathbb{R}\right)\right)$,

$$\begin{array}{c}\hfill P_{\eta }h\left(x\right):={\int }_{\mathbb{R}}h\left(y\right)p_{\eta }(x,y)dy.\end{array}$$

Let us introduce now TV norm ${\parallel ·\parallel }_{TV}$ and TV distance $d_{TV}$ respectively. Let $(X,\mathfrak{X})$ be a measurable space. Consider ${\mathbb{F}}_b$ the set of all bounded and measurable functions from $(X,\mathfrak{X})$ to $\mathbb{R}$. For any finite signed measure $\xi$ on $(X,\mathfrak{X})$, the total variation norm of $\xi$ is defined by

$${\parallel \xi \parallel }_{TV}:=sup\left\{\xi \left(f\right)|f\in {\mathbb{F}}_b,{\left|f\right|}_{\infty }⩽1\right\},$$

where ${\left|f\right|}_{\infty }:=sup\left\{\left|f\right(x\left)\right||x\in X\right\}$. For any two probability measures $\xi$ and ${\xi }^{\prime }$ on $(X,\mathfrak{X})$, the total variation distance between $\xi$ and ${\xi }^{\prime }$ is defined by

$$d_{TV}(\xi ,{\xi }^{\prime }):={\displaystyle \frac{1}{2}}{\parallel \xi -{\xi }^{\prime }\parallel }_{TV}.$$

(8)

Throughout the paper, sometimes supplied with some indices, $\tau$ and c denote positive constants, whose value may vary from line to line. Subsequent symbols with the prefix $\tau$ will be redefined according to specific scenarios and are independent of each other. For any $(a,b)\in {\mathbb{R}}^2$, $a\vee b=max\{a,b\}$ and $a\wedge b=min\{a,b\}$. When we write “C is an ($m,\epsilon \nu$)-small set”, it will be always assumed that $\epsilon \in (0,1]$ and $\nu$ is a probability measure.

The paper is organized as follows. Section 2 derives some basic properties (Proposition 1) and existence and uniqueness of the invariant probability measure ${\pi }_{\eta }$ (Theorem 1) of the EM scheme ${\left({\theta }_k\right)}_{k⩾0}$. Section 3 obtains the convergence rate of the chain ${\left({\theta }_k\right)}_{k⩾0}$ towards ${\pi }_{\eta }$ under TV distance (Theorem 2), through demonstrating furthermore the equivalent conditions for its uniform geometric ergodicity (Proposition 2). To prove Proposition 2, we use the splitting construction approach, which is described in Section 4. In Section 5, we discuss some connections and special highlights of the present work with classical ergodic theory for Markov chains on general state space, and some future research directions. In the Appendix, some known and useful properties of non-atomic and atomic Markov kernels are collected respectively in Appendix A and Appendix B. In Appendix C, we state some auxiliary results.

2. Existence and Uniqueness of Invariant Probability Measure ${\mathit{\pi }}_{\mathit{\eta }}$

Consider the EM scheme ${\left({\theta }_n\right)}_{n⩾0}$ defined by (6) with Markov kernel $P_{\eta }$ given by (7). In the sequel, let ${\mathbb{P}}_x$ denote the probability measure on the canonical space $\left({\mathbb{R}}^{\mathbb{N}},{\left(\mathcal{B}\left(\mathbb{R}\right)\right)}^{\otimes \mathbb{N}}\right)$ of the chain ${\left({\theta }_n\right)}_{n⩾0}$, with the initial state ${\theta }_0=x$. And ${\mathbb{E}}_x$ denotes the expectation under the probability measure ${\mathbb{P}}_x$.

Hypothesis (H2).

For any $x\in \mathbb{R}$ and $\eta \in (0,1)$, $|x+\eta g(x\left)\right|⩽c$.

We have the following basic property about the Markov chain ${\left({\theta }_n\right)}_{n⩾0}$ and its kernel $P_{\eta }$. Proposition 1 (1) below provides a sufficient condition for the existence of accessible $(1,\mu )$-small set C satisfying $\mu \left(C\right)>0$.

Proposition 1.

(1) 

If C is a compact subset of $\mathbb{R}$ such that $Leb\left(C\right)>0$, then C is an accessible $(1,ϵ\nu )$-small set, where

$$\begin{array}{c}\hfill ϵ:={\displaystyle \frac{Leb\left(C\right)}{\sqrt{\eta }\sigma }}\underset{(x,y)\in C^2}{inf}\varphi \left({\displaystyle \frac{y-x-\eta g\left(x\right)}{\sqrt{\eta }\sigma }}\right),\end{array}$$

(9)

$\nu (·):=Leb(·\cap C)/Leb\left(C\right)$, and $\varphi \left(x\right):={\displaystyle \frac{1}{\sqrt{2\pi }}}exp\{-x^2/2\}$ is the probability density function of standard normal distribution.

(2) 

For any $\eta \in (0,1)$, the Markov kernel $P_{\eta }$ is irreducible and strongly aperiodic.

(3) 

Under Hypothesis (H2), the state space $\mathbb{R}$ of the Markov chain ${\left({\theta }_k\right)}_{k⩾0}$ is an 1-small set.

Proof. 

(1)

Suppose that C is a compact subset of $\mathbb{R}$ such that $\mathrm{Leb}\left(C\right)>0$. Recall that $P_{\eta }$ admits a Markov kernel $p_{\eta }(x,y)$, given by

$$p_{\eta }(x,y)={\displaystyle \frac{1}{\sqrt{\eta }\sigma }}\varphi \left({\displaystyle \frac{y-x-\eta g\left(x\right)}{\sqrt{\eta }\sigma }}\right).$$

Then for all $x\in C$ and $A\in \mathcal{B}\left(\mathbb{R}\right)$,

$$\begin{array}{c}\hfill P_{\eta }(x,A)={\int }_A{p}_{\eta }(x,y)dy⩾{\int }_{A\cap C}{p}_{\eta }(x,y)dy⩾ϵ{\displaystyle \frac{\mathrm{Leb}(A\cap C)}{\mathrm{Leb}\left(C\right)}},\end{array}$$

where $ϵ\in (0,1]$ is defined in (9). Evidently, C is an $(1,ϵ\nu )$-small set, with $\nu (·)=\mathrm{Leb}(·\cap C)/\mathrm{Leb}\left(C\right)$. Furthermore, C is accessible since for all $x\in \mathbb{R}$,

$$P_{\eta }(x,C)={\displaystyle \frac{1}{\sqrt{\eta }\sigma }}{\int }_C\varphi \left({\displaystyle \frac{y-x-\eta g\left(x\right)}{\sqrt{\eta }\sigma }}\right)dy>0.$$

(2)

We have proved in (1) that any compact subset C of $\mathbb{R}$ satisfying $\mathrm{Leb}\left(C\right)>0$ is an accessible $(1,\mu )$-small set with $\mu \left(C\right)=ϵ\nu \left(C\right)>0$. Hence, $P_{\eta }$ is irreducible and strongly aperiodic.

(3)

Assume Hypothesis (H2). Then we have that for a given non-empty set $A\in \mathcal{B}\left(\mathbb{R}\right)$, both ${inf}_{x\in \mathbb{R}}(x+\eta g\left(x\right))$ and ${sup}_{x\in \mathbb{R}}(x+\eta g\left(x\right))$ exist in $\mathbb{R}$; for a given $y\in A$,

$$\underset{x\in \mathbb{R}}{inf}{p}_{\eta }(x,y)⩾{\displaystyle \frac{1}{\sqrt{2\pi \eta }\sigma }}exp\left\{-{\displaystyle \frac{1}{2\eta {\sigma }^2}}\left[{\left(y-\underset{x\in \mathbb{R}}{inf}(x+\eta g\left(x\right))\right)}^2\vee {\left(y-\underset{x\in \mathbb{R}}{sup}(x+\eta g\left(x\right))\right)}^2\right]\right\}>0.$$

Let $f\left(y\right)$ denote the function in the middle of the two inequalities above. Then for all $x\in \mathbb{R}$ and $A\in \mathcal{B}\left(\mathbb{R}\right)$,

$$P_{\eta }(x,A)={\int }_A{p}_{\eta }(x,y)dy⩾\mu \left(A\right),$$

where $\mu \left(A\right)={\int }_{A}f\left(y\right)dy$ is a Borel measure defined on $\mathcal{B}\left(\mathbb{R}\right)$. Therefore, the state space $\mathbb{R}$ is an $(1,\mu )$-small set.

□

If C is a subset of $\mathbb{R}$, let ${\tau }_C$ and ${\sigma }_C$ denote respectively the first hitting time and the first return time of C for the chain ${\left({\theta }_n\right)}_{n⩾0}$, i.e.,

$${\tau }_C=inf\{n⩾0|{\theta }_n\in C\},$$

$${\sigma }_C=inf\{n⩾1|{\theta }_n\in C\}.$$

For a given $\eta \in (0,1)$, set

$$b_{\eta }:=2\left(g^2\left(0\right)-L^2\right){\eta }^2+\left({\displaystyle \frac{1}{2}}{K}_1+{\sigma }^2+2c\right)\eta ,$$

(10)

where L, $K_1$ and c are given from (2), (3) and (5) respectively. Consider a function $\lambda \left(\eta \right)$ defined by

$$\lambda \left(\eta \right):=1-{\displaystyle \frac{1}{2}}{K}_1\eta +2L^2{\eta }^2,\eta \in (0,1).$$

By Lemma A6 (1) in Appendix C, there exists a constant ${\eta }_0\in (0,1)$ depending only on $K_1$ and L, such that for any $\eta \in (0,{\eta }_0]$, $0<\lambda \left(\eta \right)<1$. So by (10), we have for $\eta \in (0,{\eta }_0]$,

$$b_{\eta }=1-\lambda \left(\eta \right)+2g^2\left(0\right){\eta }^2+{\sigma }^2\eta +2c\eta >0.$$

(11)

For $\eta \in (0,{\eta }_0]$, consider the sets defined as follows

$$D_{\eta }:=\left\{x\in \mathbb{R}\left|\right|x|⩽{\displaystyle \frac{2b_{\eta }}{K_1{\eta }^2}}\right\}$$

and

$$B_{\eta }:=\left\{x\in \mathbb{R}\left|\right|x|⩽{\displaystyle \frac{2b_{\eta }}{K_1\eta }}\right\}.$$

Define a function $V\left(x\right)$ by

$$V\left(x\right):=1+x^2,x\in \mathbb{R}.$$

We have the following lemma. The inequality (12) below shows that for any initial state $x\in \mathbb{R}$, the quantity ${\mathbb{E}}_x\left({\beta }_{\eta }^{{\sigma }_{D_{\eta }}}\right)$ is bounded from above uniformly in $\eta$.

Lemma 1.

Under Hypothesis (H1), the following results hold.

(1) 

For any $\eta \in (0,1)$, $D_{\eta }$ is an accessible $(1,\mu )$-small set with $\mu \left(D_{\eta }\right)>0$.

(2) 

There exists a constant ${\eta }_0\in (0,1)$ depending only on $K_1$ and L, such that for any $\eta \in (0,{\eta }_0]$ and $x\in \mathbb{R}$,

$${\mathbb{E}}_x\left({\beta }_{\eta }^{{\sigma }_{D_{\eta }}}\right)⩽V\left(x\right)+b_{\eta }{\beta }_{\eta },$$

where ${\beta }_{\eta }>1$ is a constant depending on η.

Furthermore, for any $\eta \in (0,{\eta }_0]$ and $x\in \mathbb{R}$,

$${\mathbb{E}}_x\left({\beta }_{\eta }^{{\sigma }_{D_{\eta }}}\right)⩽{\mathbb{E}}_x\left({\beta }_0^{{\sigma }_{D_0}}\right)⩽V\left(x\right)+b_0{\beta }_0,$$

(12)

where $D_0:=D_{{\eta }_0}$, ${\beta }_0:={\beta }_{{\eta }_0}$ and $b_0:=b_{{\eta }_0}$.

Proof. 

(1)

Since for any $\eta \in (0,1)$, $D_{\eta }$ is a compact subset of $\mathbb{R}$ and $\mathrm{Leb}\left(D_{\eta }\right)={\displaystyle \frac{4b_{\eta }}{K_1{\eta }^2}}>0$, from Proposition 1 (1), we have that for any $\eta \in (0,1)$, $D_{\eta }$ is an accessible $(1,\mu )$-small set, with $\mu =ϵ\nu$, $ϵ\in (0,1]$ and $\nu \left(D_{\eta }\right)=1>0$, so that $\mu \left(D_{\eta }\right)>0$.

(2)

Using (5), (6) and the first inequality of (4), we can prove (see also (A.2) in [16]) that for any $\eta \in (0,1)$ and $x\in \mathbb{R}$, the following inequality holds

$$P_{\eta }V\left(x\right)⩽\lambda \left(\eta \right)V\left(x\right)+b_{\eta }{\mathbb{1}}_{B_{\eta }}\left(x\right).$$

(13)

By (11), we have ${\displaystyle \frac{b_{\eta }}{K_1\eta }}<{\displaystyle \frac{b_{\eta }}{K_1{\eta }^2}}$, for $\eta \in (0,{\eta }_0]$. Consequently, $B_{\eta }\subset D_{\eta }$ and ${\mathbb{1}}_{B_{\eta }}⩽{\mathbb{1}}_{D_{\eta }}$. From (13), we obtain

$$P_{\eta }V\left(x\right)⩽\lambda \left(\eta \right)V\left(x\right)+b_{\eta }{\mathbb{1}}_{D_{\eta }}\left(x\right).$$

Then according to Proposition 4.3.3 (ii) in [17], we have for any $\eta \in (0,{\eta }_0]$ and $x\in \mathbb{R}$,

$$\begin{array}{c}\hfill {\mathbb{E}}_x\left({\beta }_{\eta }^{{\sigma }_{D_{\eta }}}\right)⩽V\left(x\right)+b_{\eta }{\beta }_{\eta },\end{array}$$

(14)

where ${\beta }_{\eta }=1/\lambda \left(\eta \right)>1$.

Applying Lemma A6 (1) in Appendix C again, since $\lambda \left(\eta \right)$ is decreasing in $\eta$, we have for any $\eta \in (0,{\eta }_0]$, ${\beta }_{\eta }⩽{\beta }_0$; from Lemma A6 (2) in Appendix C, $D_0\subset D_{\eta }$, so ${\displaystyle \underset{\eta \in (0,{\eta }_0]}{max}{\sigma }_{D_{\eta }}⩽{\sigma }_{D_0}}$. Therefore, by (14), we come to the result (12).

□

Using the lemma above, we can obtain the following theorem, which shows the convergence of the kernel $P_{\eta }$ to its invariant probability measure ${\pi }_{\eta }$ under TV distance, for any initial probability measure $\xi$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$.

Theorem 1.

Under Hypothesis (H1), there exists a constant ${\eta }_0\in (0,1)$ depending only on $K_1$ and L, such that for any $\eta \in (0,{\eta }_0]$, the Markov kernel $P_{\eta }$ has a unique invariant probability measure ${\pi }_{\eta }$. Furthermore, there exist constants $\delta >1$, $\tau <\infty$, ${\beta }_0>1$ and an accessible $(1,\mu )$-small set $D_0$ (all depending only on $K_1$ and L) with $\mu \left(D_0\right)>0$, such that

$$\underset{x\in D_0}{sup}{\mathbb{E}}_x\left({\beta }_0^{{\sigma }_{D_0}}\right)<\infty ,$$

(15)

and for any initial probability measure ξ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$,

$${\displaystyle \sum _{n=1}^{+\infty }}{\delta }^n{d}_{TV}(\xi P_{\eta }^n,{\pi }_{\eta })⩽\tau {\mathbb{E}}_{\xi }\left({\beta }_0^{{\sigma }_{D_0}}\right).$$

(16)

Proof. 

From Lemma 1, we have that $D_0$ is an accessible $(1,\mu )$-small set with $\mu \left(D_0\right)>0$ and there exists ${\beta }_0>1$ such that (15) holds. Applying Theorem 11.4.2 from [17] to the accessible $(1,\epsilon \nu )$-small set $D_0$, we can obtain immediately the existence and uniqueness of the invariant probability measure ${\pi }_{\eta }$ for the Markov kernel $P_{\eta }$ and there exist constants $\delta >1$ and $\tau <\infty$ (both depending only on $K_1$ and L) such that for any initial probability measure $\xi$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$, (16) holds. □

Example 1

(AR(1) process). For the existence and uniqueness of invariant probability measure in Theorem 1, one can take a simple example as follows. Let $g\left(x\right):=\left(1-{\displaystyle \frac{1}{\eta }}\right)x$, for $x\in \mathbb{R}$; and $\sigma =1$. We can see that Hypothesis (H1) is satisfied in this case. The model (6) becomes

$${\theta }_{k+1}=\eta {\theta }_k+\sqrt{\eta }{\epsilon }_{k+1},k⩾0,$$

(17)

which is an $AR\left(1\right)$ model. Iterating (17), we have

$${\theta }_k={\eta }^k{\theta }_0+A_k,k⩾1,$$

where $A_k:=\sqrt{\eta }{\sum }_{i=0}^{k-1}{\eta }^i{\epsilon }_{k-i}$. Let ${\epsilon }_0$ be a standard normal random variable independent with ${\left({\epsilon }_k\right)}_{k⩾1}$. Consider the random variable $B_k:=\sqrt{\eta }{\sum }_{i=0}^{k-1}{\eta }^i{\epsilon }_i$, for any $k⩾1$. Since $0<\eta <1$, using martingale convergence theorem, we can obtain ${lim}_{k\to +\infty }{B}_k=B_{\infty }$ a.s., where $B_{\infty }:=\sqrt{\eta }{\sum }_{i=0}^{+\infty }{\eta }^i{\epsilon }_i$. Let ${\pi }_{\eta }$ be the probability distribution of $B_{\infty }$. Then we have that ${\pi }_{\eta }$ is the unique invariant probability measure for $P_{\eta }$.

Remark 1.

Theorem 1 improves the result in Lemma 2.3 of [16] in the following two aspects. It shows not only the existence of invariant probability measure ${\pi }_{\eta }$, but also the uniqueness under the same conditions, by introducing the accessible small set $D_0$. It establishes furthermore the geometric rate of convergence of $\xi P_{\eta }^n$ towards ${\pi }_{\eta }$ under TV distance for any initial probability measure ξ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$, as $n\to \infty$. See also the corollary below.

Corollary 1.

Under Hypothesis (H1), there exist constants $\delta >1$ and ${\eta }_0\in (0,1)$ depending only on $K_1$ and L, such that for any initial probability measure ξ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$ and $\eta \in (0,{\eta }_0]$,

$$d_{TV}\left(\xi P_{\eta }^n,{\pi }_{\eta }\right)=o\left({\delta }^{-n}\right),asn\to \infty .$$

3. Geometric Rate of Convergence and Uniform Geometric Ergodicity of the Kernel ${\mathit{P}}_{\mathit{\eta }}$

Let P be a Markov kernel on $X\times \mathcal{X}$. The Markov kernel P is said to satisfy the geometric drift condition (see also Definition 14.1.5, [17]) $D_g(V,\lambda ,b)$, if $V:X\to [1,\infty )$ is a measurable function, $\lambda \in [0,1)$, $b\in [0,\infty )$ and

$$PV\left(x\right)⩽\lambda V\left(x\right)+b{\mathbb{1}}_X\left(x\right).$$

(18)

We obtain the following geometric rate of convergence for the Markov kernel $P_{\eta }$ to its invariant probability measure ${\pi }_{\eta }$, for any initial state $x\in \mathbb{R}$.

Theorem 2.

Under Hypotheses (H1) and (H2), for any $\eta \in (0,1)$, there exist $\delta \in (1,\infty ]$ and $\tau <\infty$ such that for any $n\in \mathbb{N}$,

$$\underset{x\in \mathbb{R}}{sup}{∥P_{\eta }^n(x,·)-{\pi }_{\eta }∥}_{TV}⩽\tau {\delta }^{-n}$$

(19)

and

$$\underset{x\in \mathbb{R}}{sup}{d}_{TV}\left(P_{\eta }^n(x,·),{\pi }_{\eta }\right)⩽\left(\tau /2\right){\delta }^{-n}.$$

(20)

To prove the theorem above, we will need the following proposition, which gives equivalent conditions for uniform geometric ergodicity of $P_{\eta }$.

Proposition 2.

For any $\eta \in (0,1)$, the following statements are equivalent.

(1) 

$P_{\eta }$ is uniformly geometrically ergodic, i.e., $P_{\eta }$ admits an invariant probability measure ${\pi }_{\eta }$ such that there exist $\delta >1$ and $\tau <\infty$ satisfying for all $n\in \mathbb{N}$,

$$\underset{x\in \mathbb{R}}{sup}{∥P_{\eta }^n(x,·)-{\pi }_{\eta }∥}_{TV}⩽\tau {\delta }^{-n}.$$

(21)

(2) 

$P_{\eta }$ is positive, aperiodic, and there exist a small set C and a constant $\delta >1$ such that

$$\underset{x\in \mathbb{R}}{sup}{\mathbb{E}}_x\left({\delta }^{{\sigma }_C}\right)<\infty .$$

(3) 

The state space $\mathbb{R}$ is small.

Proof. 

(1) ⟹ (2). Suppose that $P_{\eta }$ is uniformly geometrically ergodic. We prove firstly that $P_{\eta }$ is irreducible; then as $P_{\eta }$ has an invariant probability measure ${\pi }_{\eta }$, $P_{\eta }$ is thus positive. Moreover, by (21), there exist $\beta >1$ and $M<\infty$ such that for any $n\in \mathbb{N}$, $x\in \mathbb{R}$ and $A\in \mathcal{B}\left(\mathbb{R}\right)$, we have

$$|P_{\eta }^n(x,A)-{\pi }_{\eta }\left(A\right)|⩽\parallel P_{\eta }^n(x,·)-{\pi }_{\eta }{\parallel }_{TV}⩽M{\beta }^{-n}.$$

(22)

Therefore, for any $n\in \mathbb{N}$, $A\in \mathcal{B}\left(\mathbb{R}\right)$ and $x\in \mathbb{R}$,

$$P_{\eta }^n(x,A)⩾{\pi }_{\eta }\left(A\right)-M{\beta }^{-n}.$$

(23)

If ${\pi }_{\eta }\left(A\right)>0$, we can choose n to be large enough such that $P_{\eta }^n(x,A)>0$, which implies that $P_{\eta }$ is irreducible.

Let C be an accessible small set. For $d>0$, define set $B_d=\{x\in \mathbb{R}|1\left(x\right)⩽d\}$. Since ${\pi }_{\eta }$ is an invariant probability measure, we get ${\pi }_{\eta }\left(C\right)>0$ and for all $x\in B_d$ we can choose n to be large enough such that

$$P_{\eta }^n(x,C)⩾{\pi }_{\eta }\left(C\right)-M{\beta }^{-n}⩾{\pi }_{\eta }\left(C\right)/2.$$

Therefore, for all $d>0$, $B_d$ is also a small set by Lemma A1, Appendix A.

Since $\mathbb{R}=\{x\in \mathbb{R}|1\left(x\right)<\infty \}$ and ${\pi }_{\eta }\left(\mathbb{R}\right)=1$, we may choose $d_0$ to be large enough so that ${\pi }_{\eta }\left(B_d\right)>0$ for all $d⩾d_0$. Since ${\pi }_{\eta }$ is an invariant probability measure of $P_{\eta }$, the set $B_d$ is accessible, for any $d⩾d_0$. Applying (23) for $A=B_d$, we may find n to be large enough such that for any $m>n$,

$$\underset{x\in B_d}{inf}{P}_{\eta }^m(x,B_d)⩾{\pi }_{\eta }\left(B_d\right)/2>0.$$

This implies that the period of $P_{\eta }$ is 1 and $P_{\eta }$ is thus aperiodic.

On the one hand, letting $A=\mathbb{R}$ in (22), one gets for any $k\in \mathbb{N}$ and $x\in \mathbb{R}$,

$$P_{\eta }^{k}1\left(x\right)⩽M{\beta }^{-k}+1.$$

(24)

We can thus choose $m\in {\mathbb{N}}^{*}$ to be large enough such that

$$M{\beta }^{-m}⩽\lambda <1.$$

As a result, $P_{\eta }^m$ satisfies the geometric drift condition $D_g(1,\lambda ,1)$, i.e., $P_{\eta }^m⩽\lambda +1$. Now, set

$$V_0\left(x\right):=1+{\lambda }^{-1/m}{P}_{\eta }1\left(x\right)+\cdots +{\lambda }^{-(m-1)/m}{P}_{\eta }^{m-1}1\left(x\right),x\in \mathbb{R}.$$

Let $b:={\lambda }^{-(m-1)/m}$. Then we have

$$\begin{array}{cc}\hfill P_{\eta }{V}_0\left(x\right)=& P_{\eta }1\left(x\right)+{\lambda }^{-1/m}{P}_{\eta }^{2}1\left(x\right)+\cdots +{\lambda }^{-(m-1)/m}{P}_{\eta }^{m}1\left(x\right)\hfill \\ \hfill ⩽& P_{\eta }1\left(x\right)+{\lambda }^{-1/m}{P}_{\eta }^{2}1\left(x\right)+\cdots +{\lambda }^{-(m-2)/m}{P}_{\eta }^{m-1}1\left(x\right)+{\lambda }^{-(m-1)/m}(\lambda +1)\hfill \\ \hfill =& {\lambda }^{1/m}({\lambda }^{-1/m}{P}_{\eta }1\left(x\right)+{\lambda }^{-2/m}{P}_{\eta }^{2}1\left(x\right)+\cdots +{\lambda }^{-(m-1)/m}{P}_{\eta }^{m-1}1\left(x\right)+1)+b\hfill \\ \hfill =& {\lambda }^{1/m}{V}_0\left(x\right)+b.\hfill \end{array}$$

(25)

On the other hand, by (24), we have for any $x\in \mathbb{R}$,

$$\begin{array}{c}\hfill 1<V_0\left(x\right)⩽\left(M{\displaystyle \sum _{k=1}^{m-1}}{\lambda }^{-k/m}{\beta }^{-k}\right)+{\displaystyle \sum _{k=0}^{m-1}}{\lambda }^{-k/m}⩽(M+1){\displaystyle \sum _{k=0}^{m-1}}{\lambda }^{-k/m}=(M+1){\displaystyle \frac{{\lambda }^{-1}-1}{{\lambda }^{-1/m}-1}}.\end{array}$$

$P_{\eta }$ thus satisfies the geometric drift condition $D_g(V_0,{\lambda }^{1/m},{\lambda }^{-(m-1)/m})$, for some $m\in {\mathbb{N}}^{*}$, $\lambda \in (0,1)$ and $V_0:\mathbb{R}\to [1,\infty )$ a measurable and bounded function on $\mathbb{R}$.

Let $f\left(x\right):=(\tilde{\lambda }-{\lambda }^{1/m})V_0\left(x\right)$, for $\tilde{\lambda }\in ({\lambda }^{1/m},1)$. From (25), for any $\tilde{\lambda }\in ({\lambda }^{1/m},1)$ and $x\in \mathbb{R}$,

$$P_{\eta }{V}_0\left(x\right)+f\left(x\right)⩽\tilde{\lambda }{V}_0\left(x\right)+b.$$

(26)

Moreover, from the above, we have for all $d>0$, the set $B_d$ is small; and for all $d⩾d_0$, the set $B_d$ becomes accessible. Therefore, these are also true for the set $\{x\in \mathbb{R}|V_0\left(x\right)⩽d\}$. Choose ${\lambda }_1\in (\tilde{\lambda },1)$. Consider a set $C:=\{x\in \mathbb{R}|V_0\left(x\right)⩽d_1\}$, where $d_1⩾d_0\vee b{({\lambda }_1-\tilde{\lambda })}^{-1}$; and hence C is accessible and small. For $x\in C$, (26) yields $P_{\eta }{V}_0\left(x\right)+f\left(x\right)<{\lambda }_1{V}_0\left(x\right)+b$. While, for $x\in C^c$, $-({\lambda }_1-\tilde{\lambda })V_0\left(x\right)<-b$ and (26) imply

$$P_{\eta }{V}_0\left(x\right)+f\left(x\right)<{\lambda }_1{V}_0\left(x\right)+b-({\lambda }_1-\tilde{\lambda })V_0\left(x\right)<{\lambda }_1{V}_0\left(x\right).$$

Equivalently, for any $x\in \mathbb{R}$,

$$P_{\eta }{V}_0\left(x\right)+f\left(x\right)⩽{\lambda }_1{V}_0\left(x\right)+b{\mathbb{1}}_C.$$

(27)

From the last inequality, with measurable functions $V_0:\mathbb{R}\to [1,\infty ]$ and $f:\mathbb{R}\to [1,\infty )$, some given $\delta ={\lambda }_1^{-1}>1$ and set C, we get for any $x\in C^c$,

$$P_{\eta }{V}_0\left(x\right)+f\left(x\right)⩽{\delta }^{-1}{V}_0\left(x\right).$$

According to (14.1.4) in Proposition 14.1.2 [17], for any $x\in \mathbb{R}$,

$${\lambda }_1^{-1}(\tilde{\lambda }-{\lambda }^{1/m}){\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\tilde{\lambda }}^{-k}{V}_0\left(X_k\right)\right)⩽{\lambda }_1^{-1}\left[P_{\eta }{V}_0\left(x\right)+f\left(x\right)\right]{\mathbb{1}}_C\left(x\right)+V_0\left(x\right){\mathbb{1}}_{C^c}\left(x\right),$$

where $0\times \infty =0$ by convention on the right-hand side of the inequality. Using (27), we obtain

$$\begin{array}{cc}\hfill {\lambda }_1^{-1}(\tilde{\lambda }-{\lambda }^{1/m}){\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k{V}_0\left(X_k\right)\right)⩽& (\underset{C}{sup}{V}_0+b{\lambda }_1^{-1}){\mathbb{1}}_C\left(x\right)+V_0\left(x\right){\mathbb{1}}_{C^c}\left(x\right)\hfill \\ \hfill ⩽& \left(d_1+b{\lambda }_1^{-1}+1\right)V_0\left(x\right)=(d_1+b\delta +1)V_0\left(x\right).\hfill \end{array}$$

Therefore, for any $x\in \mathbb{R}$,

$${\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k{V}_0\left(X_k\right)\right)⩽\tau V_0\left(x\right),$$

(28)

where $\tau :=[(d_1+1){\delta }^{-1}+b]{(\tilde{\lambda }-{\lambda }^{1/m})}^{-1}<\infty$. Moreover, for any $x\in \mathbb{R}$,

$$\begin{array}{c}\hfill {\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k{V}_0\left(X_k\right)\right)=V_0\left(x\right)+{\delta }^{-1}{\mathbb{E}}_x\left({\displaystyle \sum _{i=2}^{{\sigma }_C}}{\delta }^i{V}_0\left(X_{i-1}\right)\right)⩾1+{\delta }^{-1}{\mathbb{E}}_x\left({\delta }^{{\sigma }_C}\right).\end{array}$$

(29)

Combining (28) and (29), we have that for any $x\in \mathbb{R}$, there exists $\delta \in (1,\infty )$ such that

$${\mathbb{E}}_x\left({\delta }^{{\sigma }_C}\right)⩽\delta \left[\tau V_0\left(x\right)-1\right]<\infty .$$

(2) ⟹ (3). For any $d>0$, consider the set $D_d$ defined by

$$D_d:=\{x\in \mathbb{R}|{\mathbb{E}}_x\left({\delta }^{{\tau }_C}\right)<d\}.$$

We will firstly prove that $D_d$ is small, for any $d>0$. Note that

$$D_d=(D_d\cap C)\cup \left(D_d\cap C^c\right),$$

and the union of two small sets remains small for irreducible and aperiodic kernel $P_{\eta }$. So it suffices to prove that both $D_d\cap C$ and $D_d\cap C^c$ are small. Since C is small and $(D_d\cap C)\subseteq C$, it is evident that $D_d\cap C$ is also small. Suppose $x\in D_d\cap C^c$. By Markov’s inequality, for all $k\in {\mathbb{N}}^{*}$,

$$\begin{array}{c}\hfill {\mathbb{P}}_x({\sigma }_C⩾k+1)={\mathbb{P}}_x({\tau }_C⩾k+1)={\mathbb{P}}_x({\delta }^{{\tau }_C}⩾{\delta }^{k+1})⩽{\delta }^{-(k+1)}{\mathbb{E}}_x\left({\delta }^{{\tau }_C}\right)⩽d{\delta }^{-(k+1)}.\end{array}$$

Thus, for k that is sufficiently large,

$$\underset{x\in D_d\cap C^c}{inf}{\mathbb{P}}_x({\sigma }_C⩽k)⩾{\displaystyle \frac{1}{2}}>0.$$

Therefore, the set C is uniformly accessible from $D_d\cap C^c$. Since $P_{\eta }$ is irreducible, according to Lemma A4 (1) in Appendix A, we have that $D_d\cap C^c$ is small.

Since

$$\underset{x\in \mathbb{R}}{sup}{\mathbb{E}}_x\left({\sigma }^{{\tau }_C}\right)⩽\underset{x\in \mathbb{R}}{sup}{\mathbb{E}}_x\left({\sigma }^{{\sigma }_C}\right)<\infty ,$$

there exists $b>0$ such that $\mathbb{R}\subseteq D_b$, which is small from above. Therefore, the state space $\mathbb{R}$ is small.

(3) ⟹ (1). Suppose that the state space $\mathbb{R}$ is small set. We will firstly show that $P_{\eta }$ is irreducible, recurrent and positive. Since $\mathbb{R}$ is small, there exists $m\in \mathbb{N}$ and nonzero measure $\mu$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$ such that for all $x\in \mathbb{R}$ and $A\in \mathcal{B}\left(\mathbb{R}\right)$,

$$\begin{array}{c}\hfill P_{\eta }^m(x,A)⩾{\mu }_{\eta }\left(A\right),\end{array}$$

which implies for all $A\in \mathcal{B}\left(\mathbb{R}\right)$ such that ${\mu }_{\eta }\left(A\right)>0$, one has $P_{\eta }^m(x,A)>0$. Consequently, $P_{\eta }$ is irreducible. Since $\mathbb{R}$ is an accessible small set, ${\sigma }_{\mathbb{R}}=1$${\mathbb{P}}_x$-a.s., for all $x\in \mathbb{R}$. Applying Lemma A4 (2) in Appendix A, we have that $P_{\eta }$ is recurrent. According to Theorem 11.2.5 in [17], $P_{\eta }$ admits an invariant probability measure ${\pi }_{\eta }$ satisfying ${\pi }_{\eta }\left(C\right)<\infty$ for every accessible set C; since $\mathbb{R}$ is accessible, this implies ${\pi }_{\eta }\left(\mathbb{R}\right)<\infty$, showing that $P_{\eta }$ is positive.

Next, we will prove that $P_{\eta }$ is aperiodic by contradiction. Suppose that $P_{\eta }$ is an irreducible Markov kernel with period $d>1$. According to Lemma A3 in Appendix A, there exist $C_0$, $C_1$, ⋯, $C_{d-1}$ of mutually disjoint accessible sets such that for $i=0,\dots ,d-1$, and for any $x\in C_i$, $P_{\eta }\left(x,C_{i+1\left(\mathrm{mod}d\right)}\right)=1$ and hence

$$\begin{array}{c}\hfill P_{\eta }(x,C_{(i+1)+1\left(\mathrm{mod}d\right)})=0,\end{array}$$

(30)

where additions are in the modulo sense, and ${\bigcup }_{i=0}^{d-1}{C}_i$ is absorbing. Therefore, there exists $i_0\in \{0,\cdots ,d-1\}$ such that

$$P_{\eta }^m(x,C_{i_0})⩾{\mu }_{\eta }\left(C_{i_0}\right)>0,$$

which contradicts with (30).

Finally, we can obtain (21) by applying Proposition 3 below, with $C=\mathbb{R}$. □

Remark 2.

From the proof above, we can see that the uniform geometric ergodicity of $P_{\eta }$ is connected to geometric drift condition (18). This allows us to control the quantity in the statement (2) and thus obtain the result in the statement (3).

Proposition 3.

For any $\eta \in (0,1)$, suppose that $P_{\eta }$ admits an $(1,2\epsilon \nu )$-small set C with $\nu \left(C\right)=1$. Assume that for some $\delta >1$,

$$\begin{array}{c}\hfill \underset{x\in C}{sup}{\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C}}{\delta }^k\right)<\infty .\end{array}$$

(31)

Then there exists a constant $\beta >1$ such that for all initiate distributions ξ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$,

$$\begin{array}{c}\hfill {\displaystyle \sum _{n=0}^{\infty }}{\beta }^n{\parallel \xi P_{\eta }^n-{\pi }_{\eta }\parallel }_{TV}<\infty .\end{array}$$

(32)

The proof of the proposition above will be given in the next section.

Proof of Theorem 2.

The inequality (19) is an immediate consequence of Proposition 1 (3) and Proposition 2 (3). And the inequality (20) is obtained from (19) and the definition of TV distance (8). □

4. Splitting Construction and Split Markov Kernel ${\check{\mathit{P}}}_{\mathit{\eta }}$

Let $\check{X}:=\mathbb{R}\times \{0,1\}$ and $(\check{X},\check{\mathfrak{X}})$ be a measurable space. In this section, based on the kernel $P_{\eta }$, we will construct a new Markov kernel ${\check{P}}_{\eta }$ on the extended state space $(\check{X},\check{\mathfrak{X}})$. The detailed method of splitting construction for more general case can be found in Section 11.1 of [17].

Without loss of generality, we assume that $P_{\eta }$ admits an $(1,2\epsilon \nu )$-small set C with $\epsilon \in (0,1)$ and $\nu \left(C\right)=1$. Let $b_{\epsilon }$ be the Bernoulli distribution with success probability $\epsilon$, given by

$$b_{\epsilon }:=(1-\epsilon ){\delta }_{\left\{0\right\}}+\epsilon {\delta }_{\left\{1\right\}}.$$

For any bounded and measurable function f on $(\check{X},\check{\mathfrak{X}})$, define a function ${\overline{f}}_{\epsilon }$ on $\mathbb{R}$ by

$${\overline{f}}_{\epsilon }\left(x\right):=[{\delta }_x\otimes b_{\epsilon }]f=(1-\epsilon )f(x,0)+\epsilon f(x,1).$$

From the definition of ${\overline{f}}_{\epsilon }$ above, we have for any measure $\xi$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$,

$$\xi \left({\overline{f}}_{\epsilon }\right)=[\xi \otimes b_{\epsilon }]\left(f\right).$$

For $\eta \in (0,1)$, consider the residual kernel $R_{\eta }$ defined for $x\in \mathbb{R}$ and $A\in \mathcal{B}\left(\mathbb{R}\right)$ by

$$R_{\eta }(x,A):=\left\{\begin{array}{cc}{\displaystyle \frac{P_{\eta }(x,A)-\epsilon \nu \left(A\right)}{1-\epsilon }},\hfill & \mathrm{if}x\in C;\hfill \\ P_{\eta }(x,A),\hfill & \mathrm{if}x\notin C.\hfill \end{array}\right.$$

Now, for $\eta \in (0,1)$, define the split Markov kernel ${\check{P}}_{\eta }$ on $(\check{X},\check{\mathfrak{X}})$ as follows. For $(x,d)\in \check{X}$ and $\check{A}\in \check{\mathfrak{X}}$, set

$${\check{P}}_{\eta }(x,d;\check{A}):=Q_{\eta }(x,d;·)\otimes b_{\epsilon }\left(\check{A}\right),$$

where $Q_{\eta }$ is the Markov kernel on $\check{X}\times \mathcal{B}\left(\mathbb{R}\right)$ defined for all $B\in \mathcal{B}\left(\mathbb{R}\right)$ by

$$Q_{\eta }(x,d;B):={\mathbb{1}}_C\left(x\right)\left[{\mathbb{1}}_{\left\{0\right\}}\left(d\right)R_{\eta }(x,B)+{\mathbb{1}}_{\left\{1\right\}}\left(d\right)\nu \left(B\right)\right]+{\mathbb{1}}_{C^c}\left(x\right)P_{\eta }(x,B).$$

Equivalently, for all bounded and measurable function g on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$, one has

$$\begin{array}{cc}\hfill Q_{\eta }g(x,0)=& {\mathbb{1}}_C\left(x\right)R_{\eta }g\left(x\right)+{\mathbb{1}}_{C^c}\left(x\right)P_{\eta }g\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{{\int }_{\mathbb{R}}g\left(y\right)p_{\eta }(x,y)dy-\epsilon \nu \left(g\right)}{1-\epsilon }},\hfill & \mathrm{if}x\in C\hfill \\ {\int }_{\mathbb{R}}g\left(y\right)p_{\eta }(x,y)dy,\hfill & \mathrm{if}x\notin C\hfill \end{array}\right.;\hfill \\ \hfill Q_{\eta }g(x,1)=& {\mathbb{1}}_C\left(x\right)\nu \left(g\right)+{\mathbb{1}}_{C^c}\left(x\right)P_{\eta }g\left(x\right)=\left\{\begin{array}{cc}\nu \left(g\right),\hfill & \mathrm{if}x\in C\hfill \\ {\int }_{\mathbb{R}}g\left(y\right)p_{\eta }(x,y)dy,\hfill & \mathrm{if}x\notin C\hfill \end{array}\right.,\hfill \end{array}$$

where $p_{\eta }(x,y)={\left(2\pi \eta {\sigma }^2\right)}^{-1/2}exp\left(-{\displaystyle \frac{{(y-x-\eta g\left(x\right))}^2}{2\eta {\sigma }^2}}\right)$ is the kernel density of $P_{\eta }$. It follows that for any bounded and measurable function f on $(\check{X},\check{\mathfrak{X}})$,

$${\check{P}}_{\eta }f(x,d)=Q_{\eta }{\overline{f}}_{\epsilon }(x,d).$$

Let us describe now the canonical chain associated with the kernel ${\check{P}}_{\eta }$ on $\check{X}\times \check{\mathfrak{X}}$. For probability measure $\check{\mu }$ on $\check{\mathfrak{X}}$, denote by ${\check{\mathbb{P}}}_{\check{\mu }}$ the probability measure on the canonical space $\left({\check{X}}^{\mathbb{N}},{\check{\mathfrak{X}}}^{\otimes \mathbb{N}}\right)$ such that the coordinate process, denoted by ${\left\{({\theta }_k,D_k)\right\}}_{k⩾1}$ and called split chain, is a Markov chain with initial distribution $\check{\mu }$ and Markov kernel ${\check{P}}_{\eta }$.

From the splitting construction above, ${\left(D_n\right)}_{n⩾0}$ is a sequence of i.i.d Bernoulli random variables with success probability $\epsilon$ that is independent of ${\left({\theta }_n\right)}_{n⩾1}$. Denote by ${\left({\mathfrak{F}}_k^{\theta }\right)}_{k⩾1}$ the natural filtration of the process ${\left({\theta }_k\right)}_{k⩾1}$. An important property of the split chain ${\left\{({\theta }_k,D_k)\right\}}_{k⩾1}$ is that if ${\theta }_0$ and $D_0$ are independent, then ${\left\{({\theta }_k,{\mathfrak{F}}_k^{\theta })\right\}}_{k⩾1}$ is a Markov chain with kernel $P_{\eta }$.

For a given $\eta \in (0,1)$, suppose that $P_{\eta }$ admits an $(1,\epsilon \nu )$-small set C. We have the following lemmas: Lemmas 2–4.

Lemma 2.

For any non-negative measure ξ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$,

$$[\xi \otimes b_{\epsilon }]{\check{P}}_{\eta }^n=\xi P_{\eta }^n\otimes b_{\epsilon }.$$

Lemma 3.

For any probability measures on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$, ${\left\{({\theta }_k,{\mathfrak{F}}_k^{\theta })\right\}}_{k⩾1}$ is under ${\check{\mathbb{P}}}_{\xi \otimes b_{\epsilon }}$ a Markov chain on $\mathbb{R}\times \mathcal{B}\left(\mathbb{R}\right)$ with initial distribution ξ and Markov kernel $P_{\eta }$.

Lemma 4.

If ${\check{\lambda }}_{\eta }$ is a non-negative measure on $(\check{X},\check{\mathfrak{X}})$ and is ${\check{P}}_{\eta }$-invariant, then ${\check{\lambda }}_{\eta }={\check{\lambda }}_{\eta ,0}\otimes b_{\epsilon }$, where ${\check{\lambda }}_{\eta ,0}$ is a non-negative measure on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$, defined by

$${\check{\lambda }}_{\eta ,0}\left(A\right):={\check{\lambda }}_{\eta }(A\times \{0,1\}),A\in \mathcal{B}\left(\mathbb{R}\right).$$

In addition, ${\check{\lambda }}_{\eta ,0}$ is $P_{\eta }$-invariant.

For a given $\eta \in (0,1)$, suppose that $P_{\eta }$ admits an $(1,2\epsilon \nu )$-small set C with $\nu \left(C\right)=1$. We can obtain the following properties: Lemmas 5 and 6.

Lemma 5.

Set $\check{\alpha }:=C\times \left\{1\right\}$ and $\check{C}:=C\times \{0,1\}$. Then the following results are true:

(1) 

The set $\check{\alpha }$ is an aperiodic atom for the kernel $\check{P_{\eta }}$.

(2) 

The set $\check{C}$ is small for the kernel $\check{P_{\eta }}$.

(3) 

If C is accessible, then the atom $\check{\alpha }$ is accessible for $\check{P_{\eta }}$, and hence ${\check{P}}_{\eta }$ is irreducible.

(4) 

For all $k⩾1$, ${\check{P}}_{\eta }^k\left(\check{\alpha },\check{\alpha }\right)=\epsilon \nu P_{\eta }^{k-1}\left(C\right)$.

(5) 

If C is Harris recurrent for $P_{\eta }$, then for any probability measure ξ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$ satisfying ${\mathbb{P}}_{\xi }({\sigma }_C<\infty )=1$, ${\check{\mathbb{P}}}_{\xi \otimes {\delta }_d}({\sigma }_{\check{\alpha }}<\infty )=1$ for all $d\in \{0,1\}$. Moreover, if $P_{\eta }$ is Harris recurrent, then ${\check{P}}_{\eta }$ is Harris recurrent.

(6) 

If C is accessible and $P_{\eta }$ admits an invariant probability measure ${\pi }_{\eta }$, then $\check{\alpha }$ is positive for ${\check{P}}_{\eta }$.

Lemma 6.

Assume that for some $\delta >1$,

$$\begin{array}{c}\hfill \underset{x\in C}{sup}{\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C}}{\delta }^k\right)<\infty .\end{array}$$

(33)

Then:

(1) 

There exist constants $\gamma \in (1,\delta )$ and $\tau <{\tau }_1<\infty$ such that

$$\begin{array}{c}\hfill \underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}}}{\gamma }^k\right)⩽\tau \underset{x\in C}{sup}{\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k\right),\end{array}$$

(34)

and for any non-negative measure ξ on $(\mathbb{R},\mathcal{B}(\mathbb{R}\left)\right)$,

$$\begin{array}{c}\hfill {\check{\mathbb{E}}}_{\xi \otimes b_{\epsilon }}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}}}{\gamma }^k\right)⩽{\tau }_1{\mathbb{E}}_{\xi }\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k\right);\end{array}$$

(35)

(2) 

${\check{P}}_{\eta }$ admits a unique invariant probability measure ${\pi }_{\eta }\otimes b_{\epsilon }$, where ${\pi }_{\eta }$ is the unique invariant probability measure of $P_{\eta }$.

Proof. 

(1)

The condition (33) implies that

$$\underset{x\in C}{inf}{\mathbb{P}}_x({\sigma }_C<\infty )=1,$$

so the set C is Harris recurrent (see also Proposition 4.2.5 (ii), [17]). By Lemma 5 (5), for all $(x,d)\in \check{C}$, ${\check{\mathbb{P}}}_{(x,d)}({\sigma }_{\check{C}}<\infty )=1$ and ${\check{\mathbb{P}}}_{(x,d)}({\sigma }_{\check{\alpha }}<\infty )=1$. So we have for all $(x,d)\in \check{C}$ and $\gamma \in (1,\delta )$,

$$\begin{array}{cc}\hfill {\check{\mathbb{E}}}_{(x,d)}\left({\gamma }^{{\sigma }_{\check{\alpha }}}\right)=& \gamma {\check{\mathbb{E}}}_{(x,d)}\left({\gamma }^{{\sigma }_{\check{\alpha }}-1}\right)\hfill \\ \hfill ⩽& \gamma {\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}-1}}{\gamma }^k\right),\hfill \end{array}$$

(36)

which implies that for any $(x,d)\in \check{C}$ and $\gamma \in (1,\delta )$,

$$\begin{array}{c}\hfill {\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}}}{\gamma }^k\right)⩽(\gamma +1){\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}-1}}{\gamma }^k\right).\end{array}$$

(37)

On the other hand, for any $x\in C$, we have by Lemma 3,

$$\begin{array}{c}\hfill {\check{\mathbb{E}}}_{{\delta }_x\otimes b_{\epsilon }}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{C}}-1}}{\delta }^k\right)={\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k\right).\end{array}$$

(38)

Note that for any positive random variable Y,

$$\underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left(Y\right)⩽{\tau }_{\epsilon }\underset{x\in C}{sup}{\check{\mathbb{E}}}_{{\delta }_x\otimes b_{\epsilon }}\left(Y\right),$$

with ${\tau }_{\epsilon }:={\epsilon }^{-1}\vee {(1-\epsilon )}^{-1}$. Applying the above inequality to $Y={\sum }_{k=0}^{{\sigma }_{\check{C}}-1}{\delta }^k$, and combining (38) and the condition (33), we get

$$\begin{array}{cc}\hfill \underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{C}}-1}}{\delta }^k\right)⩽{\tau }_{\epsilon }\underset{x\in C}{sup}{\check{\mathbb{E}}}_{{\delta }_x\otimes b_{\epsilon }}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{C}}-1}}{\delta }^k\right)=& {\tau }_{\epsilon }\underset{x\in C}{sup}{\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k\right)\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \hspace{1em}& \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}<\infty .\hfill \end{array}$$

(40)

Similar to (36), we can obtain

$$\begin{array}{c}\hfill {\check{\mathbb{E}}}_{(x,d)}\left({\delta }^{{\sigma }_{\check{C}}}\right)⩽\delta {\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{C}}-1}}{\delta }^k\right).\end{array}$$

(41)

Combining (41) and (40), we obtain

$$\begin{array}{c}\hfill \underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left({\delta }^{{\sigma }_{\check{C}}}\right)<\infty .\end{array}$$

By Lemma 5 (5), ${inf}_{(x,d)\in \check{C}}{\check{\mathbb{P}}}_{(x,d)}(X_1\in \check{\alpha })>0$. Applying Theorem 14.2.3 in [17] with $A=\check{C}$, $B=\check{\alpha }$, $h=1$ and $q=1$, and using (39), we get there exist $\gamma \in (1,\delta )$, ${\tau }_0<\infty$ such that

$$\begin{array}{cc}\hfill \underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}-1}}{\gamma }^k\right)⩽& {\tau }_0\underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{C}}-1}}{\delta }^k\right)\hfill \\ \hfill ⩽& {\tau }_0{\tau }_{\epsilon }\underset{x\in C}{sup}{\mathbb{E}}_x\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\delta }^k\right).\hfill \end{array}$$

Combining the last inequality with (37), the inequality (34) is thus obtained with $\tau :=(\gamma +1){\tau }_0{\tau }_{\epsilon }<\infty$.

Consider the shift operator T, defined by $T({\omega }_0,{\omega }_1\dots ):=({\omega }_1,{\omega }_2\dots )$, for any $\omega =({\omega }_0,{\omega }_1\dots )\in {\mathbb{R}}^{\mathbb{N}}$. Define inductively $T_0$ as the identity function, i.e., $T_0\left(\omega \right)=\omega$, for $\omega \in {\mathbb{R}}^{\mathbb{N}}$; and $T_n=T_{n-1}\circ T$, for $n⩾1$. Notice that ${\sigma }_{\check{\alpha }}⩽{\sigma }_{\check{C}}+{\sigma }_{\check{\alpha }}\circ T_{{\sigma }_{\check{C}}}$ on the event $\{{\sigma }_{\check{C}}<\infty \}$ and using Lemma 3, we get

$$\begin{array}{cc}\hfill {\check{\mathbb{E}}}_{\xi \otimes b_{\epsilon }}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}}}{\gamma }^k\right)⩽& {\check{\mathbb{E}}}_{\xi \otimes b_{\epsilon }}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{C}}-1}}{\gamma }^k\right)+{\check{\mathbb{E}}}_{\xi \otimes b_{\epsilon }}\left({\displaystyle \sum _{k={\sigma }_{\check{C}}}^{{\sigma }_{\check{\alpha }\circ T_{{\sigma }_{\check{C}}}}}}{\gamma }^k\right)\hfill \\ \hfill ⩽& {\check{\mathbb{E}}}_{\xi \otimes b_{\epsilon }}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{C}}-1}}{\gamma }^k\right)+{\check{\mathbb{E}}}_{\xi \otimes b_{\epsilon }}\left({\gamma }^{{\sigma }_{\check{C}}}\right)\underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}}}{\gamma }^k\right)\hfill \\ \hfill ⩽& {\mathbb{E}}_{\xi }\left({\displaystyle \sum _{k=0}^{{\sigma }_C-1}}{\gamma }^k\right)\left[1+\gamma \underset{(x,d)\in \check{C}}{sup}{\check{\mathbb{E}}}_{(x,d)}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}}}{\gamma }^k\right)\right].\hfill \end{array}$$

Therefore, by (34) and (33), there exists ${\tau }_1\in (\tau ,\infty )$ such that the inequality (35) is satisfied.

(2)

Let $\eta \in (0,1)$ be fixed. From Proposition 1 (2), $P_{\eta }$ is irreducible and aperiodic. By Lemma A2 in Appendix A and the condition (33), the small set C is accessible. The condition (33) implies that ${sup}_{x\in C}{\mathbb{E}}_x\left({\sigma }_C\right)<\infty$. From Lemma A4 (3) in Appendix A, we have $P_{\eta }$ is positive. Thus by Lemma 5, $\check{\alpha }$ is accessible, aperiodic, and positive atom for the split kernel ${\check{P}}_{\eta }$. Using the inequality (34), the condition (33) implies that there exists $\gamma \in (1,\delta )$ such that

$$\begin{array}{c}\hfill {\check{\mathbb{E}}}_{\check{\alpha }}\left({\displaystyle \sum _{k=0}^{{\sigma }_{\check{\alpha }}}}{\gamma }^k\right)<\infty .\end{array}$$

(42)

Applying Theorem 11.4.2 in [17] to ${\check{P}}_{\eta }$, we get it has a unique invariant probability measure ${\check{\pi }}_{\eta }$, which is expressed as ${\pi }_{\eta }\otimes b_{\epsilon }$ by Lemma 4, where ${\pi }_{\eta }$ is the unique invariant probability measure for $P_{\eta }$.

□

• Using the lemmas above, we can prove Proposition 3 as follows.

Proof of Proposition 3.

Lemma 2 implies for $n⩾1$,

$$\parallel \xi P_{\eta }^n-{\pi }_{\eta }{\parallel }_{TV}⩽{\parallel (\xi \otimes b_{\epsilon }){\check{P}}_{\eta }^n-{\pi }_{\eta }\otimes b_{\epsilon }\parallel }_{TV}.$$

So we have

$$\begin{array}{c}\hfill {\displaystyle \sum _{n=1}^{\infty }}{\beta }^n\parallel \xi P_{\eta }^n-{\pi }_{\eta }{\parallel }_{TV}⩽{\displaystyle \sum _{n=1}^{\infty }}{\beta }^n{\parallel (\xi \otimes b_{\epsilon }){\check{P}}_{\eta }^n-{\pi }_{\eta }\otimes b_{\epsilon }\parallel }_{TV}.\end{array}$$

(43)

In Lemma 6, it is proven that $\check{\alpha }$ is an accessible, aperiodic, and positive atom for ${\check{P}}_{\eta }$, which admits unique invariant probability measure ${\pi }_{\eta }\otimes b_{\epsilon }$; moreover, it is proven that (42) is true. Now, applying Lemma A5 in Appendix B to ${\check{P}}_{\eta }$ with $\alpha =\check{\alpha }$, we obtain that there exist $\beta \in (1,\gamma )$ and $\tau <\infty$ such that

$${\displaystyle \sum _{n=1}^{\infty }}{\beta }^n{\parallel (\xi \otimes b_{\epsilon }){\check{P}}_{\eta }^n-{\pi }_{\eta }\otimes b_{\epsilon }\parallel }_{TV}⩽\tau {\check{\mathbb{E}}}_{\xi \otimes b_{\epsilon }}\left({\displaystyle \sum _{n=1}^{{\sigma }_{\check{\alpha }}}}{\gamma }^n\right).$$

Combining this inequality with (35) and (43) in Lemma 6 and the condition (33), we can obtain the desired inequality (32). □

5. Conclusions

This work investigates the convergence rate of EM scheme ${\left({\theta }_k\right)}_{k⩾0}$ to the invariant measure under total variation distance for solution approximations of SDE given by (1). To this end, the ergodic properties of the EM scheme, as a non-atomic Markov chain, are studied. It turns out that under Hypothesis (H1), ${\left({\theta }_k\right)}_{k⩾0}$ is irreducible, strongly aperiodic (see also Proposition 1), and admits a unique invariant probability measure ${\pi }_{\eta }$ (see also Theorem 1). If in addition Hypothesis (H2) is satisfied, ${\left({\theta }_k\right)}_{k⩾0}$ is uniformly geometrically ergodic, and converges to ${\pi }_{\eta }$ under TV distance at least in geometric rate, which is independent of the step size $\eta$ (see also Theorem 2). To show the uniform geometric ergodicity of the chain, we studied its equivalent conditions (see also Proposition 2) with the approach of splitting construction.

Let us describe some connections and special highlights of the present work with classical ergodic theory for Markov chains on general state space. In the book of Meyn and Tweedie [18], the theory on geometric ergodicity for $\psi$-irreducible Markov chain, with Markov kernel P and state space X, is introduced in Chapter 15. Our geometric ergodicity results in Theorem 1 are similar to those under f-norm in Theorem 15.4.1 of [18], when $f={\mathbb{1}}_X$. Their results hold under the following “Minorization Condition” of the chain: for some $\delta >0$, some $C\in \mathcal{B}\left(X\right)$ and some probability measure $\nu$ with $\nu \left(C^c\right)=0$ and $\nu \left(C\right)=1$,

$$P(x,A)⩾\delta {\mathbb{1}}_C\left(x\right)\nu \left(A\right),A\in \mathcal{B}\left(X\right),x\in X.$$

Meanwhile, our geometric ergodicity results in Theorem 1, obtained by using Theorem 11.4.2 in [17], only require the condition of existence of an $(1,\epsilon \nu )$-small set D, with $\epsilon \in (0,1)$ and $\nu \left(D\right)=1$, which is ensured in our case because it is shown in Proposition 3.1 (3) that the state space is small. In this article, we applied a new splitting construction approach introduced by Dedecker and Gouëzel [19], which is different from the two classical splitting approaches in [18]: Nummelin splitting technique, due to [20], and random renewal time approach, due to [21]. Thanks to this new splitting construction, it makes our proof of the equivalent conditions for the uniform geometric ergodicity of $P_{\eta }$ in Proposition 2 differ from the proof of the existing similar results in Theorems 16.0.2 and 16.2.2 of [18]. Moreover, due to this new approach, we have managed to establish an important link between 1-geometrically recurrent small set as described in (31) and the convergence of the series involving TV norm in (32) for the Markov kernel $P_{\eta }$.

With the new concepts related to non-atomic Markov chains, our results provide further properties for an EM scheme with constant step size of SDEs. In the future, EM scheme with non-constant step sizes and algorithm with high performance such as convergence rate in a controlled manner, can be considered for applications. Also, the convergence behavior of the EM scheme in higher-dimensional spaces under different distances can be considered as a future research direction.