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 C by both processes. Under the condition of existence a dominating sequence for both renewal sequences generated by the chains and non-lattice condition for renewal probabilities an upper bound for the expectation of the simultaneous renewal time is obtained.
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 C by both chains at the same time.
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 [21] or [23]. At the same time, the problem under consideration is important from both theoretical and practical perspectives. Some of the existing results can be found in the paper [8], dedicated to investigating continuous-time renewal processes generated by a sequence of independent but not necessary identically distributed random variables. The paper shows that a general renewal equation is solvable using numerical methods and demonstrates an application of this result in actuarial science. Another paper related to the renewal theory (of geometrically ergodic time-homogeneous Markov chains) is [2].
Simultaneous renewal plays an essential role in investigation of the stability of Markov chains, particularly in the time-inhomogeneous case. In the papers [9, 10] a particular coupling construction has been introduced to derive a stability estimate. A key point of the whole stability research is to construct a stochastic dominating sequence for a series of coupling times and estimate the coupling time expectation using that sequence. Later, the stability estimates obtained in [17] were applied to the widow pension actuarial model which allows calculating variables related to this actuarial schema, such as net-premiums and others. In the papers [15, 16, 19, 20] stability estimates are obtained for discrete-space Markov chains in both time-homogeneous and time-inhomogeneous case using the so-called Maximal Coupling technique.
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 [4, 5]. In [10] we have extended results from [4] and [5] to the stability of two different (but close in some sense) time-inhomogeneous Markov chains by introducing a modified coupling technique which allows chains to couple and de-couple. Such construction leads us to the necessity of investigating properties of the renewal proceses generated by time-inhomogeneous Markov chains. Many works of other authors are using similar coupling techniques to obtain stability estimates or convergence rates, see for example [1, 7] where subgeometrical Markov chains are studied.
Simultaneous renewal has been studied in other works as well. In the article [18] an estimate for a coupling moment (what we call simultaneous renewal in the present article) for time-homogeneous processes was obtained under the existence condition of the second moment of the renewal sequence. The critical feature of the proof there was the Daley inequality (see [3]) which gives an estimate for average time of the excess of a renewal process in the homogeneous case. Later, in the paper [13] the Daley inequality was extended to the time-inhomogeneous case which allows obtaining an estimate for simultaneous renewal in the case of a square-integrable dominating sequence for a time-inhomogeneous process (see [12]). This estimate involves the second moment of the domination sequence, and in the present paper, we relax that condition and only require the existence of the first moment. We will compare the results of the present paper and [12].
In the paper [14] there is a theorem that shows finiteness of the expectation of the simultaneous renewal time but a computable estimate is not available. There are also examples of how to check the non-lattice regularity condition for a renewal sequence which is involved in all the results related to the time-inhomogeneous case.
We also ask the reader to pay attention to the classical books [22, 24], devoted to the coupling method.
This paper consists of 4 sections. Section 1 contains introduction and main definitions.
The main result is stated in Section 2. It also includes a detailed explanation of the conditions of the main theorem.
An example of an application of the main result concerning a birth-death process can be found in Section 3.
Section 4 is just the conclusion.
Definitions and notations
The main object of the study is a pair of time-inhomogeneous, independent, discrete time Markov chains defined on the probability space (Ω,F,P), with the values in the general state space (E,ℰ). We will denote these chains as Xn(1) and Xn(2), n≥0, respectively.
We will use the following notation for the one-step transition probabilities
Plt(x,A)=P{Xt+1(l)∈A|Xt(l)=x},
for any x∈E, l∈{1,2}, and any A∈ℰ.
Our primary goal is to obtain an upper bound for the expectation of simultaneous hitting a given set C by both chains. An application usually dictates Applications usually dictate the choice of the set C. For a discrete state space consisting of integers the very typical choice is C={0}, although this is not a single possible choice. Moreover, we do not assume that the set C consists of a single element. We will also allow arbitrary initial conditions for the chains, and assume that each chain has a finite expectation for a first time hitting the set C for a given initial state.
We will continue to use the definitions and notations originated in [14] and [12].
Define the renewal intervals
θ0(l)=inf{t≥0,Xt(l)∈C},θn(l)=inf{t>θn−1(l),Xt(l)∈C}−θn−1(l),{\theta _{n-1}^{(l)}},{X_{t}^{(l)}}\in C\big\}-{\theta _{n-1}^{(l)}},\end{array}\]]]>
where l∈{1,2}, n≥1, and renewal times
τn(l)=∑k=0nθk(l),l∈{1,2},n≥0.
Then, for each x∈C, we can define the renewal probabilities
gn(t,l)(x)=P{θk(l)=n|τk−1(l)=t,Xt(l)=x},l∈{1,2},n≥1,
and the sequence
gn(t,l)=supx∈Cgn(t,l)(x),n≥1.
We will also need a notation for sums of gn(t,l):
Gn(t,l)=∑k>ngk(t,l).n}{g_{k}^{(t,l)}}.\]]]>Gn(t,l) can be interpreted as an estimate for the probability that renewal has not happened up to the time n.
The time of the simultaneous hitting the set C is defined as
T=inf{t>0:∃m,n,t=τm(1)=τn(2)}.0:\hspace{2.5pt}\exists m,n,\hspace{2.5pt}t={\tau _{m}^{(1)}}={\tau _{n}^{(2)}}\big\}.\]]]>
Main result
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.
Condition A
There is a non-increasing sequence Gn,n≥0, such that:
Gn(t,l)≤Gn,∑n≥0Gn=m<∞.
To make the further reasoning simpler, we allow the index n in Gn to be negative. In this case we assume: Gn=G0, n<0.
We do not require this dominating sequence to be probabilistic and allow G0≥1. Condition A will enable us to ascertain that both chains X(1),X(2) will have a finite expectation of the renewal time, which is a necessary condition for having a finite expectation of simultaneous renewal.
Opposing the paper [12] we do not require the existence of the second moment for a dominating sequence.
Next, we need some regularity condition on un(t,l) which guarantees its separation from 0.
Condition B
There is a constant γ>00$]]> and a number n0≥0, such that for all t≥0, l∈{1,2}, and n≥n0P{Xn+t(l)∈C|Xn(l)∈C}≥γ.
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 [11], Theorems 4.1, 4.2, 4.3. We will use some of them later.
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 [12]:
ν0:=min{j≥1:τj1>n0},B0:=τν01,ν1:=min{j≥ν0:τj2−τν01>n0,orτj2−τν01=0},B1:=τν12−τν01,{n_{0}}\big\},\\ {} \displaystyle {B_{0}}:={\tau _{{\nu _{0}}}^{1}},\\ {} \displaystyle {\nu _{1}}:=\min \big\{j\ge {\nu _{0}}:{\tau _{j}^{2}}-{\tau _{{\nu _{0}}}^{1}}>{n_{0}},\hspace{2.5pt}\text{or}\hspace{2.5pt}{\tau _{j}^{2}}-{\tau _{{\nu _{0}}}^{1}}=0\big\},\\ {} \displaystyle {B_{1}}:={\tau _{{\nu _{1}}}^{2}}-{\tau _{{\nu _{0}}}^{1}},\end{array}\]]]>
and further on,
ν2m:=min{j≥ν2m−1:τj1−τν2m−12>n0,orτj1−τν2m−12=0},B2m:=τν2m1−τν2m−12,ν2m+1:=min{j≥ν2m:τj2−τν2m1>n0,orτj2−τν2m=01,B2m+1:=τν2m+12−τν2m1.{n_{0}},\hspace{2.5pt}\text{or}\hspace{2.5pt}{\tau _{j}^{1}}-{\tau _{{\nu _{2m-1}}}^{2}}=0\big\},\\ {} \displaystyle {B_{2m}}:={\tau _{{\nu _{2m}}}^{1}}-{\tau _{{\nu _{2m-1}}}^{2}},\\ {} \displaystyle {\nu _{2m+1}}:=\min \{j\ge {\nu _{2m}}:\hspace{2.5pt}{\tau _{j}^{2}}-{\tau _{{\nu _{2m}}}^{1}}>{n_{0}}\hspace{0.1667em},\hspace{2.5pt}\text{or}\hspace{2.5pt}{\tau _{j}^{2}}-{\tau _{{\nu _{2m}}=0}^{1}},\\ {} \displaystyle {B_{2m+1}}:={\tau _{{\nu _{2m+1}}}^{2}}-{\tau _{{\nu _{2m}}}^{1}}.\end{array}\]]]>
We will call νk renewal trials. Let’s define τ=min{n≥1:Bn=0} and put Bt=0, if t≥τ.
For ease of use, we will introduce the variable
Sn=∑k=0nBk.
Note, that S2n=τν2n1, S2n+1=τν2n+12, however, the use of the variable Sn makes the text much more readable.
Assume that chains(Xn(l)),l∈{1,2}, start with some initial distributionsλ1andλ2and that expectations of the first time each process hits the set C are finite. Denote them asm1(λ1)andm2(λ2)respectively. Assume also, that conditions (A) and (B) introduced above hold true. In this case, there exists a dominating sequenceSˆnfor the simultaneous renewal time T:P(T>n)≤Sˆn,n)\le {\hat{S}_{n}},\]]]>such thatE[T]≤∑n≥0Sˆn≤m1(λ1)+m2(λ2)+(n0G0+m)(1+γ)/γ.
First, we assume that θ0(2)=0, which means that the second chain starts from the set C.
Next, the upper bound for T was introduced in [12] (it trivially follows from definitions):
T≤θ0(1)+∑n=0τBn=θ0(1)+∑n≥0Bn1τ>n.n}}.\]]]>
We introduce the variable
T′=∑n=0τBn.
Then θ0(1)+T′ is pointwise bigger than T, and so it stochastically dominates T.
So we will focus on building a dominating sequence for T′ and estimating its expectation.
Let us consider
P{T′>n}=∑k=0nP{Sk≤n<Sk+1}=∑k=0n∑j=knP{Sk=j,Bk+1>n−j}.n\big\}={\sum \limits_{k=0}^{n}}\mathbb{P}\{{S_{k}}\le n<{S_{k+1}}\}={\sum \limits_{k=0}^{n}}{\sum \limits_{j=k}^{n}}\mathbb{P}\{{S_{k}}=j,{B_{k+1}}>n-j\}.\]]]>
Note that some terms are equal to 0 in the latter sum because each renewal trial takes at least n0 steps, but this fact does not affect our reasoning.
Lemma 1 gives us the inequality
P{Sk=j,Bk+1>n−j}≤P{Sk=j,τ≥k}Gn−j−n0,n-j\}\le \mathbb{P}\{{S_{k}}=j,\tau \ge k\}{G_{n-j-{n_{0}}}},\]]]>
which allows us to build the upper bound for P{T′>n}n\}$]]>:
P{T′>n}≤∑k=0n∑j=knP{Sk=j,τ≥k}Gn−j−n0=∑j=0nGn−j−n0∑k=0jP(Sk=j,τ≥k).n\big\}\le {\sum \limits_{k=0}^{n}}{\sum \limits_{j=k}^{n}}\mathbb{P}\{{S_{k}}=j,\tau \ge k\}{G_{n-j-{n_{0}}}}\\ {} & \hspace{1em}={\sum \limits_{j=0}^{n}}{G_{n-j-{n_{0}}}}{\sum \limits_{k=0}^{j}}P({S_{k}}=j,\tau \ge k).\end{aligned}\]]]>
So we can put
Sˆn′=∑j=0nGn−j−n0∑k=0jP(Sk=j,τ≥k),
and so the dominating sequence for P{T′>n}n\}$]]> is built.
Lemma 8.5 from [12] gives us the inequality
P{τ>n}≤(1−γ)nn\}\le {(1-\gamma )^{n}}\]]]>
which yields
E[T′]≤(n0G0+m)(1+1/γ)=(n0G0+m)(1+γ)/γ.
Now, we have to get rid of the assumption θ0(2)=0. Following the same calculations as in [14] after formula (20) we may estimate T:
T≤θ0(1)+θ0(2)+T′,
which entails the existence of a dominating sequence Sˆn≥P{T>n}n\}$]]> and the estimate
E[T]≤∑n≥0Sˆn≤m1(λ1)+m2(λ2)+(n0G0+m)(1+γ)/γ.
□
Assume that the chainX(2)starts fromx∈C. Then, for allk,j,t≥0the following inequality holds true:P{Sk=j,Bk+1>t}≤P(Sk=j,τ≥k)Gt−n0.t\}\le P({S_{k}}=j,\tau \ge k){G_{t-{n_{0}}}}.\]]]>
We start with the assumption that k is even. We can write then:
P{Sk=j,Bk+1>t}=P{Sk=j,Xn(2)∉C,n∈{j,j+n0,…,j+t}}.t\}=\mathbb{P}\big\{{S_{k}}=j,{X_{n}^{(2)}}\notin C,n\in \{j,j+{n_{0}},\dots ,j+t\}\big\}.\]]]>
Let’s denote as η the last renewal of the chain X(2) before the time j+n0. By the construction of the sequence Bk,
η≥Sk−Bk=Sk−1.
Indeed, if k>00$]]>, this means that Sk−1=Sk−Bk>00$]]> is a renewal time for the chain X(2). If k=0, then we should put S−1=S0−B0=0, but the chain X(2) has started in C, so η>00$]]> by the condition of the lemma. So by construction, we have:
Sk−Bk≤η<j+n0.
Let us denote the sets
Al(s,t)={Xm(l)∉C,m∈{s,…,t}},l∈{1,2},
and inspect the sample path of the chain X(2) after the moment η:
P{Sk=j,Bk+1>t}=P{Sk=j,Xj(2)∉C,A2(j+n0,j+t)}=∑u,vP{Sk−1=u,τ≥k,A1(u+n0,j−1),Xj(1)∈C,Xv(2)∈C,A2(v+1,j+t)},t\}& =\mathbb{P}\big\{{S_{k}}=j,{X_{j}^{(2)}}\notin C,{A_{2}}(j+{n_{0}},j+t)\big\}\\ {} & =\sum \limits_{u,v}\mathbb{P}\big\{{S_{k-1}}=u,\tau \ge k,{A_{1}}(u+{n_{0}},j-1),\\ {} & \hspace{2em}\hspace{2em}{X_{j}^{(1)}}\in C,{X_{v}^{(2)}}\in C,{A_{2}}(v+1,j+t)\big\},\end{aligned}\]]]>
where u<j, and v≥u is a value of η. Note that
{Sk−1=u,Xu(1)∉C}={Sk−1=u,τ>k−1}={Sk−1=u,τ≥k}.k-1\}=\{{S_{k-1}}=u,\tau \ge k\}.\]]]>
Let us recall that the chains X(1) and X(2) are independent and so we can split the probability in the formula (12) into the product:
P{Sk−1=u,τ≥k,A1(u+n0,j−1),Xj(1)∈C,Xv(2)∈C,A2(v+1,j+t)}=∫E×CP{Sk−1=u,A1(u+n0,v),(Xv(1),Xv(2))=(dx,dy)}×P{A1(v+1,j−1),Xj(1)∈C,A2(v+1,j+t)|(Xv(1),Xv(2))=(x,y)}=∫E×CP{Sk−1=u,τ≥k,A1(u+n0,v),(Xv(1),Xv(2))=(dx,dy)}×P{A1(v+1,j−1),Xj(1)∈C|Xv(1)=x}P{A2(v+1,j+t)|Xv(2)=y}.
But P{A2(v+1,j+t)|Xv(2)=y}, y∈C, is just the probability that the next renewal after v will not happen until the moment j+t. So we have
supy∈CP{A2(v+1,j+t)|Xv(2)=y}=Gj+t−v(v,2)≤Gj+t−v,
where the latest inequality follows from the Condition A.
The above reasoning is valid in the case η<j, however, the case η∈{j+1,j+n0−1} yields the same result. The maximal possible value of v is j+n0, so, using the fact that sequence Gn is non-increasing, it follows from (13) that
supy∈CP{A2(v+1,j+t)|Xv(2)=y}≤Gj+t−v≤Gt−n0,
which gives the estimate that does not depend on v. Taking into account the inequality (14) we can derive
P{Sk=j,Bk+1>t}≤∑u,v∫E×CP{Sk−1=u,τ≥k,A1(u+n0,v),(Xv(1),Xv(2))=(dx,dy)}×P{A1(v+1,j−1),Xj(1)∈C|Xv(1)=x}Gt−n0≤P{Sk=j,τ≥k}Gt−n0.t\}\\ {} & \hspace{1em}\le \sum \limits_{u,v}{\int _{E\times C}}\mathbb{P}\big\{{S_{k-1}}=u,\tau \ge k,{A_{1}}(u+{n_{0}},v),({X_{v}^{(1)}},{X_{v}^{(2)}})=(dx,dy)\big\}\\ {} & \hspace{2em}\times \mathbb{P}\big\{{A_{1}}(v+1,j-1),{X_{j}^{(1)}}\in C|{X_{v}^{(1)}}=x\big\}{G_{t-{n_{0}}}}\le \mathbb{P}\{{S_{k}}=j,\tau \ge k\}{G_{t-{n_{0}}}}.\end{aligned}\]]]>
To complete the proof, we should consider the case when k is odd. In this case, k>00$]]> which means k−1≥0 and we don’t have a problem with k=0 like in the even case. So, similar reasoning will yield the same inequality for odd k. □
Application to the birth-death processes
In the paper [12] we had derived an estimate for a simultaneous renewal of two time-inhomogeneous birth-death processes X(1) and X(2) with the following transition probabilities on the t-th step:
Pt=αt01−αt0000…0αt101−αt10…00αt201−αt2……
and
Qt=βt01−βt0000…0βt101−βt10…00βt201−βt2……
The set C is equal to 0:
C={0}.
An estimate from [12] involves the second moment of the dominating sequence. Let us compare an estimate from [12] with the one that follows from Theorem 1.
We will start with building γ. As in [12] we notice that for every t>00$]]>, g1(l)=αt0>00$]]>, and assume
γ0=inft{αt0,βt0}>0.0.\]]]>
As it has been shown in the [12] γ can be chosen in the following way:
γ=exp(μˆln(γ0)/γ0)=γ0μˆ/γ0
with the n0=0.
We will use the same dominating sequence as in [12]. To do this, we consider a standard random walk with a parameter p>1/21/2$]]> and a probability fn of a first return to 0 in n steps. It has been shown in [6], Chapter XIII, that the generating function F(z) for fn looks like
F(z)=1−1−4p(1−p)s22(1−p).
We have shown in the [12] that Gn=∑k>nfn/pn}{f_{n}}/p$]]> is a dominating sequence (non-probabilistic) for renewal sequences generated by X(1) and X(2), if
p(1−p)≥supt,j{αtj(1−αtj),βtj(1−βtj)}
The estimate for the expectation of the simultaneous renewal time in case of both chains started at C looks like
E1=E[T]≤μˆ2/γ+μˆ1/γ2,
where
μˆ1=2/(2p−1)+1,μˆ2=(2p−1)−1(2+8(1−p)1−4p)+2/(2p−1)+1.
In our case the estimate looks like
E2=E[T]≤μˆ1(1+γ)/γ.
So, we can see that
E2=(E1−μˆ2/γ)(1+γ)γ,
for all γ∈(0,5−12), γ(1+γ)<1 which means that E2 is a better estimate than E1. In the same time, γ0 is typically a small number, which makes γ to be a very small number. So, the improvement in estimate can be significant for practical applications.
Conclusions
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 X and X′ we understand the first moment of hitting a specific set C by both chains.
We have shown that under conditions (A) and (B) an estimate has the form
E[T]≤∑n≥0Sˆn≤m1(λ1)+m2(λ2)+(n0G0+m)(1+γ)/γ.
We have shown how the parameters γ,n0,G0,m1(λ1),m2(λ2), and the sequence Sˆn included in the estimate can be calculated.
This result allows us to refine the stability estimate for two time-inhomogeneous Markov chains, which is a subject of a further investigation.
ReferencesAndrieu, C., Fort, G., Vihola, M.: Quantitative convergence rates for sugeometric Markov chains. Ann. Appl. Probab.52, 391–404 (2015) MR3372082. https://doi.org/10.1239/jap/1437658605Baxendale, P.: Renewal theory and computable convergence rates for geometrically ergodic Markov chains. Ann. Appl. Probab.15, 700–738 (2005) MR2114987. https://doi.org/10.1214/105051604000000710Daley, D.: Tight bounds for the renewal function of a random walk. Ann. Probab.8, 615–621 (1980) MR0573298Douc, R., Moulines, E., Soulier, P.: Practical drift conditions for subgeometric rates of convergence. Ann. Appl. Probab.14, 1353–1377 (2004) MR2071426. https://doi.org/10.1214/105051604000000323Douc, R., Moulines, E., Soulier, P.: Quantitative bounds on convergence of time-inhomogeneous Markov chains. Ann. Appl. Probab.14, 1643–1665 (2004) MR2099647. https://doi.org/10.1214/105051604000000620Feller, W.: An Introduction to Probability Theory and Its Applications, Vol. 1. John Wiley and Sons (1957) MR0088081Fort, G., Roberts, G.O.: Subgeometric ergodicity of strong Markov processes. Ann. Appl. Probab.15, 1565–1589 (2005) MR2134115. https://doi.org/10.1214/105051605000000115Gismondi, F., Janssen, J., R., M.: Non-homogeneous time convolutions, renewal processes and age-dependent mean number of notorcar accidents. Ann. Actuar. Sci.9, 36–57 (2015)Golomoziy, V.: A subgeometric estimate of the stability for time-homogeneous Markov chains. Theory Probab. Math. Stat.81, 35–50 (2010) MR2667308. https://doi.org/10.1090/S0094-9000-2010-00808-8Golomoziy, V.: An estimate of the stability for nonhomogeneous Markov chains under classical minorization condition. Theory Probab. Math. Stat.88, 35–49 (2014) MR3112633. https://doi.org/10.1090/S0094-9000-2014-00917-5Golomoziy, V.: An inequality for the coupling moment in the case of two inhomogeneous Markov chains. Theory Probab. Math. Stat.90, 43–56 (2015) MR3241859. https://doi.org/10.1090/tpms/948Golomoziy, V.: An estimate for an expectation of the simultaneous renewal for time-inhomogeneous Markov chains. Mod. Stoch. Theory Appl.315–323 (2016) MR3593115. https://doi.org/10.15559/16-VMSTA68Golomoziy, V.: An estimate of the expectation of the excess of a renewal sequence generated by a time-inhomogeneous Markov chain if a square-integrable majorizing squence exists. Theory Probab. Math. Stat.94, 53–62 (2017) MR3553453. https://doi.org/10.1090/tpms/1008Golomoziy, V., Kartashov, M.: On the integrability of the coupling moment for time-inhomogeneous Markov chains. Theory Probab. Math. Stat.89, 1–12 (2014) MR3235170. https://doi.org/10.1090/S0094-9000-2015-00930-3Golomoziy, V., Kartashov, M.: Maximal coupling and stability of discrete non-homogeneous Markov chains. Theory Probab. Math. Stat.91, 17–27 (2015) MR2986452. https://doi.org/10.1090/S0094-9000-2013-00891-6Golomoziy, V., Kartashov, M.: Maxmimal coupling and v-stability of discrete nonhomogeneous Markov chains. Theory Probab. Math. Stat.93, 19–31 (2016) MR3553437. https://doi.org/10.1090/tpms/992Golomoziy, V., Kartashov, M., Kartashov, Y.: Impact of the stress factor on the price of widow’s pensions. proofs. Theory Probab. Math. Stat.92, 17–22 (2016) MR3330687Kartashov, M., Golomoziy, V.: Average coupling time for independent discrete renewal processes. Theory Probab. Math. Stat.84, 77–83 (2011) MR2857418. https://doi.org/10.1090/S0094-9000-2012-00855-7Kartashov, M., Golomoziy, V.: Maximal coupling procedure and stability of discrete Markov chains. i. Theory Probab. Math. Stat.86, 93–104 (2013) MR3241447. https://doi.org/10.1090/S0094-9000-2014-00905-9Kartashov, M., Golomoziy, V.: Maximal coupling procedure and stability of discrete Markov chains. ii. Theory Probab. Math. Stat.87, 65–78 (2013) MR3241447. https://doi.org/10.1090/S0094-9000-2014-00905-9Kluppelberg, C., S., P.: Renewal theory for functionals of a Markov chain with compact state space. Ann. Probab.3, 2270–2300 (2003) MR2016619. https://doi.org/10.1214/aop/1068646385Lindvall, T.: Lectures on Coupling Method. John Wiley and Sons (1991) MR1180522Melfi, V.: Nonlinear Markov renewal theory with statistical applications. Ann. Probab.20, 753–771 (1992) MR1159572Thorisson, H.: Coupling, Stationarity, and Regeneration. Springer, New York (2000) MR1741181. https://doi.org/10.1007/978-1-4612-1236-2