Maximal Coupling Procedure and Stability of Continuous-time Markov Chains

In this study we first investigate the stability of subsampled discrete Markov chains through the use of the maximal coupling procedure. This is an extension of the available results on Markov chains and is realized through the analysis of the subsampled chain , where is an increasing sequence of random stopping times. Then the similar results are realized for the stability of countable-state Continuous-time Markov processes by employing the skeleton-chain method.


Introduction
Loosely speaking, coupling may be referred to as the practice of constructing two(or more) probability measures(or random processes) within a single measurable space for the purposes of investigating similarities and individual characteristics of each of the coupled measures. Coupling techniques which include maximal coupling(also known as Vasershtein coupling) and other forms of coupling like weak coupling(often referred to as distributional coupling) amongst others have had a very profound influence in the studies of stability and ergodicity of Markov processes. Of particular interest to our study is the maximal coupling technique which appeared in Griffeath [5] and has since been employed in its various forms by many researchers amongst them Connor [2], Roberts and Rosenthal [10] and Kartashov and Golomozyi [7]. We first discuss discrete Markov chains(DTMCs) and important information about coupling in general before going into the maximal coupling specifics and later countable-state Continuous-time Markov processes.
We let , then we denote by a DTMC on a state space with q-matrix denotes the к-th power of this matrix. We also let be the product of the measure  and the matrix Q. The transition function of the DTMC is denoted by , where here and hereafter 1 C is the indicator function of set C and P i and E i respectively denote the probability and expectation of the chain under the condition that . For a chain with 1-step transition probabilities P′ the similar notation of P′ i and E′ i respectively will be used for the chain Ф′ n .
Let T be the random time such that for . Further let be the coupling of Ф n and Ф′ n , then for the coupling time T we get the coupling inequality (1) where D is a sequence of random variables on {0,1} (independent of both Ф n and Ф′ n ), and the variable d n is set equal to 0 starting from T.
Coupling is said to be successful if is called the coupling event. We note that chain Ф n is said to be weakly ergodic if , where is the Dirac point mass at i. Which implies that Ф n is weakly ergodic if for any pair of initial states successful coupling exists. Maximal coupling is achieved when we get  (1). The interested reader can consult Lindvall [8] for more study on coupling.
Kartashov and Golomozy [7] studied the stability of distributions of discrete chains under small perturbations of 1-step transition probabilities, then proved for an arbitrary number of steps, the stability for the transition probabilities. This was achieved through the use of the maximal coupling technique. In this study, we prove that the stability holds not only for discrete Markov chains as in [7], but also for the subsampled chain , where is a non-decreasing sequence of stopping times, with . Then prove that the same results hold for Continuous-time Markov chains.
The paper is organized as follows. In Section 2, we present some minor results similar to previous contributions on stability with focus on the stability of Discrete-time Markov chains under maximal coupling and the results of [7] and related references will play a pivotal role here. The main results of our study are availed in Section 3. In Section 4 we have the conclusions. The Appendix is the 5th Section which contains the proofs.

Theorem 1. [Theorem 4 in [5]]
Any Markov chain Ф n has a maximal coupling, that is, it achieves equality in formula (1). Thus the maximal coupling is successful if and only if the chain Ф n is weakly ergodic.
As noted in [7], at each 1-step transition time, for maximal coupling the chains evolve coupled with maximal probability of which is equal to the weight of the maximal common component of the corresponding transition distributions and the joint evolution distributions. Further the chains evolve independently of each other with probability 1-a. According to Theorem 1 we proceed with the rest of the study with the understanding that existence of maximal coupling is a given. Assumption 2 below also reinforces the fact that irrespective of the initial state coupling occurs almost surely. We desire to prove in this Section that the stability holds for the subsampled chain , where is an increasing sequence of random stopping times. We expect the distance to be a decreasing function of . We also note that , for any h>0 thus i is an atom and the singleton set {i} is a small set for the chain Ф nh . Thus some small set C exists for the chain Ф nh . Further we conclude that all the assumptions that we had for the discrete case also hold for the chain Ф nh .

Proposition 1.
Let all the conditions of Theorem 2 hold then for any and for some (and then for all)stopping times we have,

Proof
See Proof 2 in the Appendix.
From Proposition 1 it is obvious that the rest of the DTMCs results like Theorem 3 follow in a similar manner, hence the omission of such results. Furthermore we give the following results which are closely related to the CTMCs in this study.

V-norm Stability
The results that follow stated in the form of Propositions are focused on the V-norm. We did not assume in Assumptions 3 that the function is bounded either on or on the small set C. In the CTMCs results below we assume . Further we are satisfied to confirm similarities of our results with the results of authors like [1], [7]  If we have a continuously differentiable increasing concave function with then we have a subgeometric drift inequality for the first inequality of (7).

Proposition 4. Let
and be CTMCs and let the measurable functions be such that . Then for some positive constant  there exists a function such that for any two initial states i, k and for the increasing sequence of stopping times we have (9) Proof See Proof 4 in the Appendix.
For the sake of consistency we note that inequality (9) still holds for the norm . The following Proposition 5 confirms the consistency of the Propositions 3 and 4. It generalizes Theorem 3.6 of [1]. We define the pair of inverse Young functions . All pairs , satisfy and . Let be a subgeometric rate function. For more on subgeometric rate functions the interested reader can consult [9]. We note that if then is also a subgeometric rate function and so is provided for some constant .

Proposition 5.
Let and be aperiodic and irreducible CTMCs and let the measurable functions , be such that and . Then there exists some positive constant  * such that for the pair of inverse Young functions , any two initial states i, k, and for the increasing sequence of stopping times we have (10)

Conclusions
We note that the CTMCs results in this study hold also for , where are continuous-time processes in the space of cadlag functions defined on with values in the Polish space , endowed with the Skorokhod topology. Also extending the results to general state Markov processes follows naturally because the coupling technique is the one tool for such an endeavor.
In the future we would like to consider 'J 1 -step uniform stability' and 'J 1 -step V -stability condition' as defined below and find out how they affect the results. Definition 1. We define the 'J 1 -step uniform stability' as follows. Instead of the 1-step probability transitions as in Assumption 1 we now assume a 1 -jump probability transition for the uniform stability condition.

Definition 2.
Similarly we define the 'J 1 -step V -stability condition' as follows. Instead of the 1step probability transitions as in Assumption 3 we now assume a 1 -jump probability transition for the V-stability condition.

BMSA Volume 10
We would like to know if the J 1 -step uniform stability and Assumption 2 hold and  <1 -, then for all jumps (11) uniformly and if this is the same as Theorem 2. Similarly we would like to know if Theorem 3 follows naturally by assuming J 1 -step V -stability condition and strong mixing condition. As already mentioned in [7], in general, the 1-step V-stability condition doesn't follow from the -step uniform stability condition, though these conditions are equivalent for V  1. Consequently, we interested at similar conclusions for the J 1 -step uniform stability and J 1 -step V-stability.
We are also interested in investigating the stability of finite-dimensional distributions instead of only the 1-dimensional distributions as done in this study.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this article.