# Subgeometric Ergodicity under Random-Time State-Dependent Drift Conditions

Authors: Mokaedi V. Lekgari
Journal: Journal of Probability and Statistics
 (2014)

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## Abstract

Motivated by possible applications of Lyapunov techniques in the stability of stochastic networks, subgeometric ergodicity of Markov chains is investigated. In a nutshell, in this study we take a look atf-ergodic general Markov chains, subgeometrically ergodic at rate r, when the random-time Foster-Lyapunov drift conditions on a set of stopping times are satisfied.


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## Body

## 1. Introduction

Ergodicity of Markov processes has, in recent years, received quite substantial attention. Of the three standard types of ergodicity, being ordinary, geometric (often referred to as exponential), and strong ergodicity, ordinary ergodicity stands out as the most well studied of the three, especially on discrete time Markov chains. However for subgeometric ergodicity, which can be referred to as a kind of convergence that is faster than ordinary ergodicity but slower than geometric ergodicity, much study is needed especially for continuous Markov processes [1].Ergodicity in the context of state-dependent drift conditions on a deterministic time index has been studied by many authors, [2–4]. Connor and Fort [2] and Yüksel and Meyn [3] studied ergodicity with a drift condition taking the form Pn(x)V(x)≤βV(x)+b1C(x) for some deterministic function n:X→[1,∞) and a constant β∈(0,1). According to Theorem 2.1(ii) of [5] a deterministic sequence of functions Vn exists V:X→[1,∞) and satisfies a Foster-Lyapunov drift condition:
(1)PVn+1≤Vn-r(n)f+br(n)1C,n∈Z+
for a petite set C∈B(X) and a constant b<∞ such that V0 is bounded on C.It has been proved that this Foster-Lyapunov condition holds not only for everyn∈Z+ but also for a sequence of stopping times {Tn}, n∈Z+ for some discrete time Markov chain (Φn)n∈Z+, [4]. The results of Zurkowski [4] relied heavily on the work of Connor and Fort [2], Meyn and Tweedie [6], and Tuominen and Tweedie [5]. The aim of this study is to investigate and refine these random-time state-dependent drift conditions results with our emphasis on the subgeometric ergodicity for a general Markov chain.As observed by Yüksel and Meyn [3], there is a function V on the state space X taking positive values and an increasing sequence of stopping times {Ti:i∈Z+}, with T0=0, such that, for each i,
(2)E[V(ΦTi+1)∣FTi]≤V(ΦTi)-δ(ΦTi),
where the function δ:X→R is positive (bounded away from zero) outside a small set and FTi denotes the filtration of events up to time Ti. The reader needs to take note that although inequality (2) is important to this study, it will not be of much use because we will only use its refinement which is availed as (23) in Theorem 7.Throughout the paper the sequence of stopping times{Ti:i∈Z+} is assumed to be nondecreasing, with T0=0. On state-dependent criteria for stability, traditionally it is assumed that the stopping times take the form
(3)Ti+1=Ti+n(Φ(Ti)),i≥0,
where n:X→N is a deterministic function of the state.Although our work will mostly be confined to working withpetite sets, we know that for ψ-irreducible and aperiodic processes petite sets are in factsmall sets [5]. Thus we need not worry that the techniques that rely on small sets in some of the previous studies may become unavailable in the random-time drift setting thinking that a small set for Φn may not necessarily be small for ΦTn. Small sets are analogous to compact sets in the stability theory of ψ-irreducible Markov chains. Also in most applications of ψ-irreducible Markov chains we find that any compact set is small.The paper is organized as follows. In Section2, we have the preliminaries which introduces basic notations, definitions, and theorems. The main results are availed in Section 3 which is mostly concerned with (f,r)-ergodicity. In Section 4 we make the conclusions of this study.


## 2. Preliminaries

LetZ+={0,1,2,…}, N+={1,2,…}, and R+=[0,∞). Let (Φn)n∈Z+(resp., (Φt)t∈R+) denote a discrete time Markov chain (DTMC), (resp., a continuous time Markov chain (CTMC) on a countable space, as given in Liu et al. [7]). The state space for all chains is denoted (X,B(X)). Unless otherwise specified ψ-irreducibility, aperiodicity, and positive recurrence of all the chains we will be dealing with is assumed. The corresponding transition function of the DTMC process is denoted by Pn(x,y)=Px(Φn=y)=Ex[1Φn=y], where 1A, here and hereafter, is the indicator function of set A and Px and Ex, respectively, denote the probability and expectation of the chain under the condition that Φ0=x. Analogously we get Pt(x,y)=Px(Φt=y)=Ex[1Φt=y] for the CTMC.


### 2.1. Subgeometric Rate Function

Let functionr∈Λ0, where Λ0 is the family of measurable increasing functions r:R+→[1,∞) satisfying log⁡r(n)/n↓0 as n↑∞. Let Λ denote the class of positive functions r¯:R+→(0,∞) such that for some r∈Λ0 we have
(4)0<lim⁡ninf⁡r¯(n)r(n)≤lim⁡nsup⁡r¯(n)r(n)<∞.
Then Λ is referred to as the class of subgeometric rate functions [8]. Indeed (4) implies the equivalence of the class of functions Λ0 with the class of functions Λ. Example of functions in the class r∈Λ is the rate r(n)=exp⁡(sn1/(1+α)), α>0, s>0, which has been discussed only recently in literature. Without loss to generality we suppose that r(0)=1 whenever r∈Λ.The properties ofr∈Λ0 which follow from (4) and are to be used frequently in this study are
(5)r(x+y)≤r(x)r(y)∀x,y∈R+(6)r(x+a)r(x)⟶1asx→∞,foreacha∈R+.


### 2.2. Subgeometric Rate Ergodicity

Letr∈Λ; then, the ergodic chain Φn is said to be subgeometric ergodic of order r in the f-norm or simply (f,r)-ergodic if for all x∈X, then
(7)lim⁡t→+∞r(n)∥Pn(x,·)-π(·)∥f=0,
where π is the unique invariant distribution of the process for a (signed) measure σ and ∥σ∥f=sup⁡|g|≤f|σ(g)| and f:X→[1,∞) is a measurable function.In the analysis of ergodicity, for any nonempty setC∈B(X) the two first hitting times are given as δC=inf⁡{n≥J1:Φn∈C}, where J1 denotes the first jump-time of the chain Φn and τC=inf⁡{n≥0:Φn∈C}. If C is a singleton consisting of state x, we write δx for δC. We also note that τC=0≠δC if Φ0∈C; however, τC=δC, if Φ0∉C. For hitting times of the continuous chain Φt we need only replace n∈Z+ with t∈R+ in the above notations and we are done. It is known that (7) holds if and only if the Foster-Lyapunov (or just Lyapunov) drift condition holds. It is for this reason that we cannot do without the Lyapunov drift conditions in our study. Let r^(t)=∫0t‍r(s)ds, for the case of Φt.


### 2.3.l-Ergodicity

Forl∈R such that l≥1, the chain Φn is called l-ergodic if max⁡x∈CEx[δCl]<∞ for some finite nonempty C⊆X.


### 2.4. Small Set

A setC∈B(X) is said to besmall (or (n,ϵ,ν)-small) if there exists some n∈Z+, ϵ>0, and a probability measure ν(·) on X such that
(8)Pn(x,·)≥ϵν(·),∀x∈C.


### 2.5. Petite Set

A setC∈B(X) is said to bepetite if there exists some nontrivial measure νa on B(X) and some probability distribution a={an:n∈Z+} such that
(9)∑n=1∞‍anPn(x,·)≥νa(·),∀x∈C.
Petite sets generalize small sets. If Ψ-irreducibility and aperiodicity of the process are assumed (as is the case throughout this study) then set C∈B+(X) is a petite set if and only if (8) is satisfied. Thus petite sets in this case are in fact small sets.Next we state conditionD and Theorems 1 and 2 which will be useful in this study especially in the proof of Theorem 8.


### 2.6. ConditionD(ϕ,V,C)

There exist a functionV:X→[1,∞), a concave monotone function ϕ:[1,∞]→(0,∞], a measurable set C, and a constant b<∞ such that
(10)PV(x)-V(x)≤ϕ∘V+b1C.Theorem 1 (Theorem14.0.1 in [6]).
If conditionD holds for some petite set C and there exists x0∈X such that V(x0)<∞, then there exists a unique invariant distribution π, π(ϕ∘V)<∞ and
(11)lim⁡n→∞∥Pn(x,·)-π(·)∥ϕ∘V=0,π–a.e.,
where ϕ∘V(x)≥1, for all x∈X.Theorem 2 (Proposition 2.5 of Douc et al. [9]).
LetP be a ψ-irreducible and aperiodic kernel. Assume that D(ϕ,V,C) holds for function ϕ with lim⁡t→+∞ϕ′(t)=0, a petite set C, and a function V with {V<+∞}≠∅. Then there exists an invariant probability measure π, and for all x in the full and absorbing set {V<∞}, we got
(12)lim⁡n→∞rϕ(n)∥Pn(x,·)-π(·)∥=0,π–a.e.,
where rϕ(n)=ϕ∘Hϕ-1(n), Hϕ(v)∶=∫1v(1/ϕ(x))dx.


## 2.1. Subgeometric Rate Function

Let functionr∈Λ0, where Λ0 is the family of measurable increasing functions r:R+→[1,∞) satisfying log⁡r(n)/n↓0 as n↑∞. Let Λ denote the class of positive functions r¯:R+→(0,∞) such that for some r∈Λ0 we have
(4)0<lim⁡ninf⁡r¯(n)r(n)≤lim⁡nsup⁡r¯(n)r(n)<∞.
Then Λ is referred to as the class of subgeometric rate functions [8]. Indeed (4) implies the equivalence of the class of functions Λ0 with the class of functions Λ. Example of functions in the class r∈Λ is the rate r(n)=exp⁡(sn1/(1+α)), α>0, s>0, which has been discussed only recently in literature. Without loss to generality we suppose that r(0)=1 whenever r∈Λ.The properties ofr∈Λ0 which follow from (4) and are to be used frequently in this study are
(5)r(x+y)≤r(x)r(y)∀x,y∈R+(6)r(x+a)r(x)⟶1asx→∞,foreacha∈R+.


## 2.2. Subgeometric Rate Ergodicity

Letr∈Λ; then, the ergodic chain Φn is said to be subgeometric ergodic of order r in the f-norm or simply (f,r)-ergodic if for all x∈X, then
(7)lim⁡t→+∞r(n)∥Pn(x,·)-π(·)∥f=0,
where π is the unique invariant distribution of the process for a (signed) measure σ and ∥σ∥f=sup⁡|g|≤f|σ(g)| and f:X→[1,∞) is a measurable function.In the analysis of ergodicity, for any nonempty setC∈B(X) the two first hitting times are given as δC=inf⁡{n≥J1:Φn∈C}, where J1 denotes the first jump-time of the chain Φn and τC=inf⁡{n≥0:Φn∈C}. If C is a singleton consisting of state x, we write δx for δC. We also note that τC=0≠δC if Φ0∈C; however, τC=δC, if Φ0∉C. For hitting times of the continuous chain Φt we need only replace n∈Z+ with t∈R+ in the above notations and we are done. It is known that (7) holds if and only if the Foster-Lyapunov (or just Lyapunov) drift condition holds. It is for this reason that we cannot do without the Lyapunov drift conditions in our study. Let r^(t)=∫0t‍r(s)ds, for the case of Φt.


## 2.3.l-Ergodicity

Forl∈R such that l≥1, the chain Φn is called l-ergodic if max⁡x∈CEx[δCl]<∞ for some finite nonempty C⊆X.


## 2.4. Small Set

A setC∈B(X) is said to besmall (or (n,ϵ,ν)-small) if there exists some n∈Z+, ϵ>0, and a probability measure ν(·) on X such that
(8)Pn(x,·)≥ϵν(·),∀x∈C.


## 2.5. Petite Set

A setC∈B(X) is said to bepetite if there exists some nontrivial measure νa on B(X) and some probability distribution a={an:n∈Z+} such that
(9)∑n=1∞‍anPn(x,·)≥νa(·),∀x∈C.
Petite sets generalize small sets. If Ψ-irreducibility and aperiodicity of the process are assumed (as is the case throughout this study) then set C∈B+(X) is a petite set if and only if (8) is satisfied. Thus petite sets in this case are in fact small sets.Next we state conditionD and Theorems 1 and 2 which will be useful in this study especially in the proof of Theorem 8.


## 2.6. ConditionD(ϕ,V,C)

There exist a functionV:X→[1,∞), a concave monotone function ϕ:[1,∞]→(0,∞], a measurable set C, and a constant b<∞ such that
(10)PV(x)-V(x)≤ϕ∘V+b1C.Theorem 1 (Theorem14.0.1 in [6]).
If conditionD holds for some petite set C and there exists x0∈X such that V(x0)<∞, then there exists a unique invariant distribution π, π(ϕ∘V)<∞ and
(11)lim⁡n→∞∥Pn(x,·)-π(·)∥ϕ∘V=0,π–a.e.,
where ϕ∘V(x)≥1, for all x∈X.Theorem 2 (Proposition 2.5 of Douc et al. [9]).
LetP be a ψ-irreducible and aperiodic kernel. Assume that D(ϕ,V,C) holds for function ϕ with lim⁡t→+∞ϕ′(t)=0, a petite set C, and a function V with {V<+∞}≠∅. Then there exists an invariant probability measure π, and for all x in the full and absorbing set {V<∞}, we got
(12)lim⁡n→∞rϕ(n)∥Pn(x,·)-π(·)∥=0,π–a.e.,
where rϕ(n)=ϕ∘Hϕ-1(n), Hϕ(v)∶=∫1v(1/ϕ(x))dx.


## 3. Main Results

The core results in this Section are presented as Theorems4, 7, and 8. Theorem 4 is a modified version of Theorem 5.1.3 in [4], while Theorem 7 is a modified version of both Theorem 2.1 in Yüksel and Meyn [3] and Proposition 5.1.2 in Zurkowski [4]. Then we have the refinement of Theorem 7 in the light of the work of [9] given as Theorem 8.The skeleton chain method will be employed in the proof of Proposition3, so next we define it. Let the chain {Φn}n∈Z+ be aperiodic and irreducible. Let h>0; for the skeleton chain Φnh we define,
(13)δ~x(h)=inf⁡{n≥1:Φnh=x};
then δx(h):=hδ~x(h) denotes the first hitting time on x of the skeleton chain Φnh. Proposition 3 will be employed in the proof of Theorem 4.Proposition 3.
For some (and equivalently for any) finite nonempty setA⊂X, some constant h>0 and any subgeometric rate function r, we have
(14)Ey[∑k=0δ~y(h)‍r(kh)]<∞.Proof.
For anyh>0, when Φ0=x, there is the possibility that the skeleton chain Φnh can miss visits of the continuous time process Φt to y resulting in δy≤δy(h). Supposing that Φ0=x, we let Dk be the kth sojourn time at y and Wk be the interval length between the kth exit from y and the next visit to y for the continuous time Markov process Φt. We note the Dk are independent and identically distributed, having an exponential distribution with parameter λ. Also we note that the Wk are independent of each other and the Dk.
We define(15)N=min⁡{Dnn≥1:theh-skeletonchainisinmmmmmiiiiiiistateyduringtheintervalDn}.
Then δy(h)≤τy+∑k=1N-1‍(Dk+Wk)+h, so by the submultiplicative property (5) we have
(16)Ex[r(δy(h))]≤r(h)Ex[r(τy)]×∑n=1∞‍Ey[r(∑k=1n-1‍(Dk+Wk))1{N=n}]≤r(h)Ex[r(τy)]×∑n=1∞‍Ey[r(∑k=1n-1‍(Dk+Wk))1⋂k=1n-1{Dk≤h}]≤r(h)Ex[r(τy)]×∑n=1∞‍Ey[r((n-1)h)r(∑k=1n-1‍Wk)](1-eλh)n-1≤r(h)Ex[r(τy)]×∑n=1∞‍r((n-1)h)(Ey[r(δy)])n-1(1-eλh)n-1.
We define D(h)=∑n=1∞‍r((n-1)h)(Ey[r(δy)])n-1(1-eλh)n-1. Also we have lim⁡h→0(1-eλh)=0; by hypothesis we get Ey[r(δy)]<∞, and by property (6) we get lim⁡n→∞(r(nh-h)/r(nh))=1; therefore we can choose h0>0 small enough such that D(h0)<∞. By (16) we get the following recursive formula:
(17)Ey[∑k=0δ~y(h0)‍r(kh0)]=Ey[r(δy(h0))]+Ey[∑k=0δ~y(h0)-1‍r(kh0)]=δ~y(h0)Ey[r(δy(h0))]+r(0)<∞.Theorem 4.
Suppose that there exist a functionV:X→(0,∞) and constants b∈R and β∈(0,1) such that for an increasing sequence of stopping times {Ti}(18)E[V(ΦTi+1)∣FTi]≤βV(ΦTi)+b1{ΦTi∈C}.
Then for any r∈Λ the chain Φn is r-ergodic.Proof.
It is worth noting thatΦTi is also an aperiodic and irreducible Markov chain that satisfies the univariate condition of (Lemma 14 of [10]); hence, V is bounded on C. For any r∈Λ and ξCz=min⁡{n≥z:Φn∈C}, with τC≤ξCz for all C∈B+(X), we have
(19)sup⁡x∈CEx[∑k=0τC-1‍r(k)]≤sup⁡x∈CEx[∑k=0δC-1‍r(k)]≤sup⁡x∈CEx[∑k=0ξCz-1‍r(k)].
Owing to the equivalence of Λ0 and Λ we suppose that r∈Λ0 therefore rate r satisfies the submultiplicative property (5). We have established by Proposition 3 that Ey[∑k=0δ~x(h)‍r(kh)]<∞ which satisfies (19) for some h=1 and so by Theorem 2.1 in [1] implies that Ey[r^(δy)]<∞. Then by Theorem 3.3 of Liu et al. [7], the set C is (1,r)-regular hence we conclude that the chain is r-ergodic.As a consequence of studying subgeometric ergodicity in the total variation norm (that is in the1-norm, when we let f≡1), it is worth noting that the results of Theorem 4 above are applicable to continuous time Markov chains, (Φt)t∈R+ on a countable state space studied in [7], by letting t=nh. Mao [11] investigated l-ergodicity when l is restricted to Z+. Liu et al. [7] extended the results to the case when l∈R+. Under random-time state-dependent drift function, we state the following corollary which is an extension of Corollary 2.1 of Liu et al. [7].Corollary 5.
Letl∈R+ with l≥1. Then the following are equivalent.(i)
The chainΦt is l-ergodic, that is, max⁡x∈CEx[δCl]<∞ for some finite nonempty set C.(ii)
For some nonempty finite setC and an increasing sequence of stopping times {Ti}, there exist ⌊l⌋+1 finite nonnegative functions Vn(ΦTi), 0≤n≤⌊l⌋ and a constant b such that V0(ΦTi)≥EΦTi[τ0l^] and
(20)∑ΦTi+1∈C‍qΦTi,ΦTi+1Vm+1(ΦTi+1)≤-(l^+m+1)Vm(ΦTi)+b1C(ΦTi)for 0≤m≤⌊l⌋-1, where l^=l-⌊l⌋ and ⌊l⌋ denote the integer-part function, that is, the largest integer which is smaller than or equal to l.Proof.
The proof is analogous to the proof of Corollary 2.1 of Liu et al [7] and follows naturally therefrom.For stochastically monotone CTMCs, some computable bounds on exponential convergence based on first hitting times and the drift function were investigated in Corollary 2.2 in Liu et al. [7] which leads us to the next corollary. The following corollary is analogous to Corollary 2.2 of Liu et al. [7] for a chain of the form (Φt)t∈R+ and also follows naturally consequent to the results of Theorem 4.Corollary 6.
LetX=Z+. Suppose that the regular and totally stable intensity matrix Q is monotone and r(t) is strictly increasing. If E0[r^(δ0)]<∞ then Φt is subgeometrically ergodic of order r and for any x∈X and any t∈R+ we have
(21)r(t)∥Pt(x,·)-π(·)∥≤Dx,
where Dx=2Ex[r(τ0)]+2Eπ[r(τ0)]<∞. In the specific case of r(t)=tl, l>1 such that l∈Z+ and the drift criterion (ii) of Corollary 5 holds for the singleton set C={0}, Vn(ΦTi), 0≤n≤⌊l⌋ and b; then, for any ΦTi=x∈X and any t∈R+(22)tl-1∥Pt(x,·)-π(·)∥≤Dx,
where Dx=2V⌊l⌋-1(x)+(2bπ0/l)≤2V⌊l⌋-1(x)+(2b/l)≤∞.Proof.
The proof is analogous to the proof of Corollary 2.2 of Liu et al. [7] and follows naturally therefrom.Theorem7 which follows deals with subgeometric rate convergence of f-ergodic chains, in the case when f isn’t a total variation norm. In a nutshell we’ll be dealing with (f,r)-regularity of the petite subset C⊂X, or simply (f,r)-ergodicity of the chain Φn.Theorem 7.
Suppose thatΦn is a ψ-irreducible and aperiodic Markov chain. Suppose further that there are functions V:X→(0,∞), δ:X→[1,∞), f:X→[1,∞), a small set C, constant b∈R and r∈Λ such that for an increasing sequence of stopping times {Ti}(23)E[V(ΦTi+1)∣FTi]≤V(ΦTi)-δ(ΦTi)+b1{ΦTi∈C};E[∑k=TiTi+1-1‍f(Φk)r(k)∣FTi]≤δ(ΦTi),i≥0
then, Φn is (f,r)-ergodic.Proof.
We define a sequence{Mi:i≥0} by
(24)Mi=V(ΦTi)-∑k=0i-1‍(δ(ΦTk)-b1{ΦTk∈C}),
where by convention ∑k=mn‍(·)=0 whenever m>n; hence, M0=V(Φ0).
Then under the assumed drift conditions we have(25)E[Mj∣FTj-1]≤V(ΦTj-1)-∑k=0j-2‍(δ(ΦTk)-b1{ΦTk∈C})=Mj-1,j≥1
which implies that {Mi:i≥0} is a supermartingale. For a measurable subset C⊂X we denote the first hitting time for the sampled chain delayed by a constant n∈Z+ as
(26)ξCn=min⁡{i≥n:ΦTi∈C};
then E[MξCn]≤M0 for any n∈Z+, and
(27)E[∑k=0ξCn-1‍δ(ΦTi)∣F0]≤M0+b<∞
because V is bounded on C.
It is assumed that(28)τC≤ξCn,∀C∈B+(X)
so that
(29)Ex[∑k=TiτC-1‍f(Φk)r(k)]≤Ex[∑k=0ξCn-1‍δ(ΦTi)]<M0+b<∞
which implies (f,r)-ergodicity of the chain Φn.Theorem8 below is similar to Proposition 5.1.4 in [4] and Theorem 2.8 in [9]. Its importance is that it generalizes Theorem 7. By employing condition D(ϕ,V,C), Theorems 1 and 2 we are a able to get quite a wide range of functions of the forms f and r.Theorem 8.
Let the chainΦn be aperiodic and irreducible, that is, both ϕ∘V-ergodic and rϕ-ergodic for some V:X→[1,∞), ϕ:[1,∞]→(1,∞] and subgeometric rate function r∈Λ. Assume that condition D(ϕ,V,C) holds for ϕ such that lim⁡t→∞ϕ′(t)=0 and a petite set C such that sup⁡C<+∞. Further let Ψi,Ψj:X→[1,∞) be pairs of ultimately nondecreasing functions, which satisfy Ψi(x)Ψj(x)≤x+y and Ψi(x),Ψj(x)→∞, where i=1 (resp., i=2) whenever j=2 (resp., j=1). Then the chain (Φn is Ψj(ϕ∘V),Ψi∘rϕ)-ergodic.Proof.
By Theorem2 we know that the chain is (1,rϕ)-ergodic and also that for a large enough x, it holds that Ψi(x),Ψj(x)≤2x. Then by Lemma 2.7 in [9] for some constant a such that Ψi(x)≤ax we have that Ψi∘rϕ∈Λ, where i/j=1,2. Hence the chain is (1,Ψi∘rϕ)-ergodic. Further Theorem 1 states that the chain is (ϕ∘V,1)-ergodic; then, it follows that there exists a class of sets where the chain is (Ψj(ϕ∘V),1)-ergodic j/i=1,2. Its worth noting that there is a balance between therate of convergence Ψi∘rϕ and thenormΨj(ϕ∘V); that is, the slower the former the larger the latter.
Because of theψ-irreducibility and aperiodicity of the chain Φn, there is only one full absorbing set S, which can be partitioned into two partitions say, Gr of (1,Ψi∘rϕ)-regular sets and say, Gn of (Ψj(ϕ∘V),1)-regular sets. The intersection of these two partitions yields the set Gr,n of (Ψj(ϕ∘V),Ψi∘rϕ)-regular sets, and by Proposition 2.6 in [9] we have
(30)sup⁡x∈Gr,nE[∑k=0τC-1Ψi(rϕ(k))Ψj(ϕ∘V(Φk))]<∞
for any accessible set C∈B+(X). Indeed (30) implies that the chain is (Ψj(ϕ∘V),Ψi∘rϕ)-ergodic.


## 4. Conclusion

The refinement of Theorem 2.1 in Yüksel and Meyn [3] and Proposition 5.1.2 in Zurkowski [4] has been accomplished in this study and presented in the form of Theorem 7. Theorems 4 and 8 follow as a consequence of Theorem 7. This study like a multitude of previous studies before it may pave way for more research on random-time state-dependent Foster-Lyapunov drift conditions especially that interest on the same has developed from researchers on control and optimization theory, information theory, and so forth.



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