The main subject of the study in this paper is the simultaneous renewal time for two time-inhomogeneous Markov chains which start with arbitrary initial distributions. By a simultaneous renewal we mean the first time of joint hitting the specific set

In this paper, we are focused on studying a simultaneous renewal time for two time-inhomogeneous, discrete-time Markov chains on a general state space. By simultaneous renewal we mean visiting some set

While general renewal theory is well developed, properties of the renewal sequences generated by time-inhomogeneous Markov chains are not studied so seriously. The literature available for this topic is scant. Some generalizations of the general homogeneous theory could be found, for example, in [

Simultaneous renewal plays an essential role in investigation of the stability of Markov chains, particularly in the time-inhomogeneous case. In the papers [

Estimates for the expectation of a simultaneous renewal time obtained in this paper and stability estimates derived from them have many practical applications. In particular, we may point out applications in the actuarial mathematics similar to the widow pension mentioned above. Other examples could be: obtaining premium in actuarial models that are represented by a perturbed Markov chain, such as seasonal factors. Another area of application is the queuing theory. Markov chains stability results enable us to study non-homogeneous qeuing models or models affected by some small non-homogeneous perturbation.

There are many works related to stability estimates which use coupling methods. For example stability estimates for different versions of the same time-homogeneous chain started with different initial distributions, using standard coupling technique could be found in the papers [

Simultaneous renewal has been studied in other works as well. In the article [

In the paper [

We also ask the reader to pay attention to the classical books [

This paper consists of 4 sections. Section

The main result is stated in Section

An example of an application of the main result concerning a birth-death process can be found in Section

Section

The main object of the study is a pair of time-inhomogeneous, independent, discrete time Markov chains defined on the probability space

We will use the following notation for the one-step transition probabilities

Our primary goal is to obtain an upper bound for the expectation of simultaneous hitting a given set

We will continue to use the definitions and notations originated in [

Define the renewal intervals

Then, for each

We will also need a notation for sums of

The time of the simultaneous hitting the set

First, we introduce a condition that guarantees finiteness of the expectation of the renewal times for both chains. In this paper, we use a dominating sequence for this purpose.

There is a non-increasing sequence

To make the further reasoning simpler, we allow the index

We do not require this dominating sequence to be probabilistic and allow

Opposing the paper [

Next, we need some regularity condition on

There is a constant

It is essential that such condition also guarantees certain “regularity” and non-periodicity of a chain. The periodic chains do not satisfy it.

There are various results which allow checking the condition (A) practically. See, for example [

Before we state the main theorem, we should introduce some definitions, which are used in the proof, and the auxiliary lemma.

The notations below are the same as in [

We will call

For ease of use, we will introduce the variable

First, we assume that

Next, the upper bound for

We introduce the variable

So we will focus on building a dominating sequence for

Let us consider

Lemma

Let’s now estimate its expectation:

Lemma 8.5 from [

Now, we have to get rid of the assumption

We start with the assumption that

The above reasoning is valid in the case

To complete the proof, we should consider the case when

In the paper [

The set

An estimate from [

We will start with building

As it has been shown in the [

We will use the same dominating sequence as in [

The estimate for the expectation of the simultaneous renewal time in case of both chains started at

So, we can see that

In this article, we obtained an estimate for the expectation of the simultaneous renewal time for two time-inhomogeneous, discrete-time Markov chains on a general state space. By the simultaneous renewal of two Markov chains

We have shown that under conditions (A) and (B) an estimate has the form

We have shown how the parameters

This result allows us to refine the stability estimate for two time-inhomogeneous Markov chains, which is a subject of a further investigation.