On the Study of Transience and Recurrence of the Markov Chain Defined by Directed Weighted Circuits Associated with a Random Walk in Fixed Environment

Abstract

By using the cycle representation theory of Markov processes, we investigate proper criterions regarding transience and recurrence of the corresponding Markov chain represented uniquely by directed cycles (especially by directed circuits) and weights of a random walk with jumps in a fixed environment.

1. Introduction

A systematic research has been developed (Kalpazidou [1], MacQueen [2], Minping and Min [3], Zemanian [4], and others) in order to investigate representations of the finite-dimensional distributions of Markov processes (with discrete or continuous parameter) having an invariant measure, as decompositions in terms of the cycle (or circuit) passage functions: 𝐽𝑐̂𝑖,𝑗=1,if𝑖,𝑗areconsecutivestatesof𝑐,̂=0,otherwise, (1) for any directed sequence 𝑐̂=(𝑖₁,𝑖₂,…,𝑖_𝑟) (or 𝑐=(𝑖₁,𝑖₂,…,𝑖_𝑟,𝑖₁)) of states called a cycle (or a circuit), 𝑟>1, of the corresponding Markov process. The representations are called cycle (or circuit) representations while the corresponding discrete parameter Markov processes generated by directed circuits 𝑐=(𝑖₁,𝑖₂,…,𝑖_𝑟,𝑖₁), 𝑟>1, are called circuit chains.

Following the context of the theory of Markov processes’ cycle-circuit representation, the present work arises as an attempt to investigate proper criterions regarding the properties of transience and recurrence of the corresponding Markov chain represented uniquely by directed cycles (especially by directed circuits) and weights of a random walk with jumps (having one elastic left barrier) in a fixed ergodic environment (Kalpazidou [1], Derriennic [5]).

The paper is organized as follows. In Section 2, we give a brief account of certain concepts of cycle representation theory of Markov processes that we will need throughout the paper. In Section 3, we present some auxiliary results in order to make the presentation of the paper more comprehensible. In particular, in Section 3, a random walk with jumps (having one elastic left barrier) in a fixed ergodic environment is considered, and the unique representations by directed cycles (especially by directed circuits) and weights of the corresponding Markov chain are investigated. These representations will give us the possibility to study proper criterions regarding transience and recurrence of the abovementioned Markov chain, as it is described in Section 4.

Throughout the paper, we will need the following notations: ℵ={0,1,2,…},ℵ^∗={},{}.1,2,…𝑍=…,−1,0,1,… (2)

2. Preliminaries

Let us consider a denumerable set S. Then the directed sequence 𝑐=(𝑖₁,𝑖₂,…,𝑖_𝑟,𝑖₁) modulo the cyclic permutations, where 𝑖₁,𝑖₂,…,𝑖_𝑟∈𝑆, 𝑟>1, completely defines a directed circuit in 𝑆. The ordered sequence 𝑐̂=(𝑖₁,𝑖₂,…,𝑖_𝑟) associated with the given directed 𝑐 is called a directed cycle in 𝑆.

A directed circuit may be considered as 𝑐=(𝑐(𝑟),𝑐(𝑟+1),…,𝑐(𝑟+𝑣−1),𝑐(𝑟+𝑣)), if there exists an 𝑟∈𝑍, such that 𝑖₁=𝑐(𝑟+0), 𝑖₂=𝑐(𝑟+1),…,𝑖_𝑣=𝑐(𝑟+𝑣−1), 𝑖₁=𝑐(𝑟+𝑣), where 𝑐 is a periodic function from 𝑍 to 𝑆.

The corresponding directed cycle is defined by the ordered sequence 𝑐̂=(𝑐(𝑟),𝑐(𝑟+1),…,𝑐(𝑟+𝑣−1)). The values 𝑐(𝑘) are the points of 𝑐, while the directed pairs (𝑐(𝑘),𝑐(𝑘+1)), 𝑘∈𝑍, are the directed edges of 𝑐.

The smallest integer 𝑝≡𝑝(𝑐)≥1 satisfying the equation 𝑐(𝑟+𝑝)=𝑐(𝑟), for all 𝑟∈𝑍, is the period of 𝑐. A directed circuit 𝑐 such that 𝑝(𝑐)=1 is called a loop. (In the present work we will use directed circuits with distinct point elements.)

Let us also consider a directed circuit 𝑐 (or a directed cycle 𝑐̂) with period 𝑝(𝑐)>1. Then we may define by 𝐽_𝑐^(𝑛)(𝑟)(),𝑖,𝑗=1,ifthereexistsan𝑟∈𝑍suchthat𝑖=𝑐,𝑗=𝑐𝑟+𝑛=0,otherwise, (3) the 𝑛-step passage function associated with the directed circuit 𝑐, for any 𝑖,𝑗∈𝑆, 𝑛≥1.

Furthermore we may define by𝐽_𝑐(𝑖)=1,ifthereexistsan𝑟∈𝑍suchthat𝑖=𝑐(𝑟),=0,otherwise, (4) the passage function associated with the directed circuit 𝑐, for any 𝑖∈𝑆. The above definitions are due to MacQueen [2] and Kalpazidou [1].

Given a denumerable set 𝑆 and an infinite denumerable class 𝐶 of overlapping directed circuits (or directed cycles) with distinct points (except for the terminals) in 𝑆 such that all the points of 𝑆 can be reached from one another following paths of circuit edges; that is, for each two distinct points 𝑖 and 𝑗 of 𝑆 there exists a finite sequence 𝑐₁,…,𝑐_𝑘, 𝑘≥1, of circuits (or cycles) of 𝐶 such that 𝑖 lies on 𝑐₁ and 𝑗 lies on 𝑐_𝑘, and any pair of consecutive circuits (𝑐_𝑛,𝑐_𝑛+1) have at least one point in common. Generally we may assume that the class 𝐶 contains, among its elements, circuits (or cycles) with period greater or equal to 2.

With each directed circuit (or directed cycle) 𝑐∈𝐶 let us associate a strictly positive weight 𝑤_𝑐 which must be independent of the choice of the representative of 𝑐, that is, it must satisfy the consistency condition 𝑤_cot𝑘=𝑤_𝑐, 𝑘∈𝑍, where 𝑡_𝑘 is the translation of length 𝑘 (that is, 𝑡_𝑘(𝑛)≡𝑛+𝑘, 𝑛∈𝑍, for any fixed 𝑘∈𝑍).

For a given class 𝐶 of overlapping directed circuits (or cycles) and for a given sequence (𝑤_𝑐)_𝑐∈𝐶 of weights we may define by 𝑝_𝑖𝑗=∑_𝑐∈𝐶𝑤_𝑐⋅𝐽_𝑐⁽¹⁾𝑖,𝑗∑_𝑐∈𝐶𝑤_𝑐⋅𝐽_𝑐(𝑖) (5) the elements of a Markov transition matrix on 𝑆, if and only if ∑_𝑐∈𝐶𝑤_𝑐⋅𝐽_𝑐(𝑖)<∞, for any 𝑖∈𝑆. This means that a given Markov transition matrix 𝑃=(𝑝_𝑖𝑗), 𝑖,𝑗∈𝑆, can be represented by directed circuits (or cycles) and weights if and only if there exists a class of overlapping directed circuits (or cycles) 𝐶 and a sequence of positive weights (𝑤_𝑐)_𝑐∈𝐶 such that the formula (5) holds. In this case, the Markov transition matrix 𝑃 has a unique stationary distribution 𝑝 which is a solution of 𝑝𝑃=𝑝 and is defined by 𝑝(𝑖)=_𝑐∈𝐶𝑤_𝑐⋅𝐽_𝑐(𝑖),𝑖∈𝑆. (6)

It is known that the following classes of Markov chains may be represented uniquely by circuits (or cycles) and weights:(i)the recurrent Markov chains (Minping and Min [3]),(ii)the reversible Markov chains.

3. Auxiliary Results

Let us consider a Markov chain (𝑋_𝑛)_𝑛≥0 on ℵ with transitions 𝑘→(𝑘+1), 𝑘→(𝑘−1), and 𝑘→𝑘 whose the elements of the corresponding Markov transition matrix are defined by 𝑃𝑋_𝑛+1=0/𝑋_𝑛=0=𝑟₀,𝑃𝑋_𝑛+1=1/𝑋_𝑛=0=𝑝₀,𝑝₀=1−𝑟₀,𝑃𝑋_𝑛+1=𝑘+1/𝑋_𝑛=𝑘=𝑝_𝑘𝑃𝑋,𝑘≥1,_𝑛+1=𝑘/𝑋_𝑛=𝑘=𝑟_𝑘𝑃𝑋,𝑘≥1,_𝑛+1=𝑘−1/𝑋_𝑛=𝑘=𝑞_𝑘,𝑘≥1, (7) such that 𝑝_𝑘+𝑞_𝑘+𝑟_𝑘=1, 0≤𝑝_𝑘, 𝑟_𝑘≤1, for every 𝑘≥1, as it is shown in (Figure 1).

Figure 1 

Assume that (𝑝_𝑘)_𝑘≥0 and (𝑟_𝑘)_𝑘≥0 are arbitrary fixed sequences with 0≤𝑝₀=1−𝑟₀≤1, 0≤𝑝_𝑘, 𝑟_𝑘≤1, for every 𝑘>1. If we consider the directed circuits 𝑐_𝑘=(𝑘,𝑘+1,𝑘), 𝑐^′_𝑘=(𝑘,𝑘),𝑘≥0, and the collections of weights (𝑤_𝑐𝑘)_𝑘≥0 and (𝑤_𝑐′𝑘)_𝑘≥0, respectively, then we may obtain that 𝑝_𝑘=𝑤_𝑐𝑘𝑤_𝑐𝑘−1+𝑤_𝑐𝑘+𝑤_𝑐′𝑘,forevery𝑘≥1, (8) with 𝑝₀=𝑤_𝑐0𝑤_𝑐0+𝑤_𝑐′0. (9)

Here the class 𝐶(𝑘) contains the directed circuits 𝑐_𝑘=(𝑘,𝑘+1,𝑘), 𝑐_𝑘−1=(𝑘−1,𝑘,𝑘−1), and 𝑐^′_𝑘=(𝑘,𝑘). Furthermore we may define 𝑞_𝑘=𝑤_𝑐𝑘−1𝑤_𝑐𝑘−1+𝑤_𝑐𝑘+𝑤_𝑐′𝑘,forevery𝑘≥1 (10) and 𝑟_𝑘=𝑤_𝑐′𝑘/(𝑤_𝑐𝑘−1+𝑤_𝑐𝑘+𝑤_𝑐′𝑘), such that 𝑝_𝑘+𝑞_𝑘+𝑟_𝑘=1, for every 𝑘≥1, with 𝑟₀=1−𝑝₀=𝑤_𝑐′0/(𝑤_𝑐0+𝑤_𝑐′0).

The transition matrix 𝑃=(𝑝_𝑖𝑗) with 𝑝_𝑖𝑗=∑^∞_𝑘=0𝑤_𝑐𝑘⋅𝐽_𝑐⁽¹⁾_𝑘𝑖,𝑗∑^∞_𝑘=0𝑤_𝑐𝑘⋅𝐽_𝑐𝑘(𝑖)+𝑤_𝑐′𝑘⋅𝐽_𝑐′𝑘(𝑖𝑝),for𝑖≠𝑗,_𝑖𝑖=∑^∞_𝑘=0𝑤_𝑐′𝑘⋅𝐽_𝑐⁽¹⁾_′𝑘(𝑖,𝑖)∑^∞_𝑘=0𝑤_𝑐𝑘⋅𝐽_𝑐𝑘(𝑖)+𝑤_𝑐′𝑘⋅𝐽_𝑐′𝑘(𝑖,) (11)(12) where 𝐽_𝑐⁽¹⁾_𝑘(𝑖,𝑗)=1, if 𝑖, 𝑗 are consecutive points of the circuit 𝑐_𝑘, 𝐽_𝑐𝑘(𝑖)=1, if 𝑖 is a point of the circuit 𝑐_𝑘, and 𝐽_𝑐′𝑘(𝑖)=1, if 𝑖 is a point of the circuit 𝑐^′_𝑘, expresses the representation of the Markov chain (𝑋_𝑛)_𝑛≥0 by directed cycles (especially by directed circuits) and weights.

Furthermore we consider also the “adjoint” Markov chain (𝑋^′_𝑛)_𝑛≥0 on ℵ whose elements of the corresponding Markov transition matrix are defined by 𝑃𝑋^′_𝑛+1=0/𝑋^′_𝑛=0=𝑟^′₀,𝑃𝑋^′_𝑛+1=1/𝑋^′_𝑛=0=𝑞^′₀,𝑞^′₀=1−𝑟^′₀,𝑃𝑋^′_𝑛+1=𝑘−1/𝑋^′_𝑛=𝑘=𝑝^′_𝑘𝑃𝑋,𝑘≥1,_′𝑛+1=𝑘/𝑋^′_𝑛=𝑘=𝑟^′_𝑘𝑃𝑋,𝑘≥1,_′𝑛+1=𝑘+1/𝑋^′_𝑛=𝑘=𝑞^′_𝑘,𝑘≥1, (13) such that 𝑝^′_𝑘+𝑞^′_𝑘+𝑟^′_𝑘=1, 0<𝑝^′_𝑘,𝑞^′_𝑘,𝑟^′_𝑘≤1, for every 𝑘≥1, as it is shown in (Figure 2).

Figure 2 

Assume that (𝑞^′_𝑘)_𝑘≥0 and (𝑟^′_𝑘)_𝑘≥0 are arbitrary fixed sequences with 0≤𝑞^′₀=1−𝑟^′₀≤1, 0≤𝑞^′_𝑘, 𝑟^′_𝑘≤1, for every 𝑘≥1. If we consider the directed circuits 𝑐_𝑘^′′=(𝑘+1,𝑘,𝑘+1), 𝑐_𝑘^′′′=(𝑘,𝑘), 𝑘≥0, and the collections of weights (𝑤_{𝑐𝑘′′})_𝑘≥0, and (𝑤_{𝑐𝑘′′′})_𝑘≥0 respectively, then we may have 𝑞^′_𝑘=𝑤_{𝑐𝑘′′}𝑤_{𝑐′′𝑘−1}+𝑤_{𝑐𝑘′′}+𝑤_{𝑐𝑘′′′},forevery𝑘≥1, (14) with 𝑞^′₀=𝑤_𝑐0′′𝑤_𝑐0′′+𝑤_{𝑐0′′′}. (15)

Here the class 𝐶^′(𝑘) contains the directed circuits 𝑐_𝑘^′′=(𝑘+1,𝑘,𝑘+1), 𝑐^′′_𝑘−1=(𝑘,𝑘−1,𝑘), and 𝑐_𝑘^′′′=(𝑘,𝑘). Furthermore we may define 𝑝^′_𝑘=𝑤_{𝑐′′𝑘−1}𝑤_𝑐^′′_𝑘−1+𝑤_{𝑐𝑘′′}+𝑤_{𝑐𝑘′′′}𝑟,forevery𝑘≥1,_′𝑘=𝑤_{𝑐𝑘′′′}𝑤_{𝑐′′𝑘−1}+𝑤_{𝑐𝑘′′}+𝑤_{𝑐𝑘′′′}, (16) such that 𝑝^′_𝑘+𝑞^′_𝑘+𝑟^′_𝑘=1, for every 𝑘≥1, with 𝑟^′₀=1−𝑞^′₀=𝑤_{𝑐0′′′}𝑤_𝑐0′′+𝑤_{𝑐0′′′}. (17)

So the transition matrix 𝑃^′=(𝑝^′_𝑖𝑗) with elements equivalent to that given by the above-mentioned formulas (11), (12) expresses also the representation of the “adjoint” Markov chain (𝑋^′_𝑛)_𝑛≥0 by directed cycles (especially by directed circuits) and weights.

Consequently we have the following.

Proposition 1. The Markov chain (𝑋_𝑛)_𝑛≥0 defined as above has a unique representation by directed cycles (especially by directed circuits) and weights.

Proof. Let us consider the set of directed circuits 𝑐_𝑘=(𝑘,𝑘+1,𝑘) and 𝑐^′_𝑘=(𝑘,𝑘), for every 𝑘≥0, since only the transitions from 𝑘 to 𝑘+1, 𝑘 to 𝑘−1, and 𝑘 to 𝑘 are possible. There are three circuits through each point 𝑘≥1, 𝑐_𝑘−1, 𝑐_𝑘, and 𝑐^′_𝑘 and two circuits through 0: 𝑐₀, 𝑐^′₀.
The problem we have to manage is the definition of the weights. We may symbolize by 𝑤_𝑘 the weight 𝑤_𝑐𝑘 of the circuit c_k and by 𝑤^′_𝑘 the weight 𝑤_𝑐′𝑘 of the circuit 𝑐^′_𝑘, for any 𝑘≥0. The sequences (𝑤_𝑘)_𝑘≥0, (𝑤^′_𝑘)_𝑘≥0 must be a solution of 𝑝_𝑘=𝑤_𝑘𝑤_𝑘−1+𝑤_𝑘+𝑤^′_𝑘,𝑘≥1with𝑝₀=𝑤₀𝑤₀+𝑤^′₀,𝑟_𝑘=𝑤^′_𝑘𝑤_𝑘−1+𝑤_𝑘+𝑤^′_𝑘,𝑘≥1with𝑟₀=𝑤^′₀𝑤₀+𝑤^′₀,𝑞_𝑘=1−𝑝_𝑘−𝑟_𝑘,𝑘≥1. (18)
Let us take by 𝑏_𝑘=𝑤_𝑘/𝑤_𝑘−1, 𝛾_𝑘=𝑤^′_𝑘/𝑤^′_𝑘−1, 𝑘≥1.
As a consequence we may have 𝑏_𝑘=𝑝_𝑘𝑞_𝑘=𝑝_𝑘1−𝑝_𝑘−𝑟_𝑘,𝛾_𝑘=𝑟_𝑘𝑟_𝑘−1𝑝_𝑘−1𝑝_𝑘𝑏_𝑘,forevery𝑘≥1. (19)
Given the sequences (𝑝_𝑘)_𝑘≥0 and (𝑟_𝑘)_𝑘≥0, it is clear that the above sequences (𝑏_𝑘)_𝑘≥1 and (𝛾_𝑘)_𝑘≥1 exist and are unique. This means that the sequences (𝑤_𝑘)_𝑘≥0 and (𝑤^′_𝑘)_𝑘≥0 are defined uniquely, up to multiplicative constant factors, by 𝑤_𝑘=𝑤₀⋅𝑏₁⋯𝑏_𝑘,𝑤^′_𝑘=𝑤^′₀⋅𝛾₁⋯𝛾_𝑘. (20)
(The unicity is understood up to the constant factors 𝑤₀, 𝑤^′₀.)

Proposition 2. The “adjoint” Markov chain (𝑋^′_𝑛)_𝑛≥0 defined as above has a unique representation by directed cycles (especially by directed circuits) and weights.

Proof. Following an analogous way of that given in the proof of Proposition 1 the problem we have also to manage here is the definition of the weights. To this direction we may symbolize by 𝑤_𝑘^′′ the weight 𝑤_𝑐^′′_𝑘 of the circuit 𝑐_𝑘^′′ and by 𝑤_𝑘^′′′ the weight 𝑤_𝑐^′′′_𝑘 of the circuit 𝑐_𝑘^′′′, for every 𝑘≥0. The sequences (𝑤_𝑘^′′)_𝑘≥0 and (𝑤_𝑘^′′′)_𝑘≥0 must be solutions of 𝑞^′_𝑘=𝑤_𝑘^′′𝑤^′′_𝑘−1+𝑤_𝑘^′′+𝑤_𝑘^′′′,𝑘≥1with𝑞^′₀=𝑤₀^′′𝑤₀^′′+𝑤₀^′′′,𝑟^′_𝑘=𝑤_𝑘^′′′𝑤^′′_𝑘−1+𝑤_𝑘^′′+𝑤_𝑘^′′′,𝑘≥1with𝑟^′₀=𝑤₀^′′′𝑤₀^′′+𝑤₀^′′′,𝑝^′_𝑘=1−𝑞^′_𝑘−𝑟^′_𝑘,𝑘≥1. (21)
By considering the sequences (𝑠_𝑘)_𝑘 and (𝑡_𝑘)_𝑘 where 𝑠_𝑘=𝑤^′′_𝑘−1/𝑤_𝑘^′′, 𝑡_𝑘=𝑤_𝑘^′′′/𝑤^′′′_𝑘−1, 𝑘≥1, we may obtain that 𝑠_𝑘=1−𝑞^′_𝑘−𝑟^′_𝑘𝑞^′_𝑘,𝑡_𝑘=𝑟^′_𝑘𝑟^′_𝑘−1⋅𝑞^′_𝑘𝑞^′_𝑘−1⋅𝑠_𝑘,forevery𝑘≥1. (22)
For given sequences (𝑞^′_𝑘)_𝑘≥0, (𝑟^′_𝑘)_𝑘≥0 it is obvious that (𝑠_𝑘)_𝑘≥1, (𝑡_𝑘)_𝑘≥1 exist and are unique for those sequences, that is, the sequences (𝑤_𝑘^′′)_𝑘≥0, (𝑤_𝑘^′′′)_𝑘≥0 are defined uniquely, up to multiplicative constant factors, by 𝑤_𝑘^′′=𝑤₀^′′𝑠₁⋅𝑠₂⋯𝑠_𝑘,𝑤_𝑘^′′′=𝑤₀^′′′⋅𝑡₁,𝑡₂⋯𝑡_𝑘. (23)
(The unicity is based to the constant factors 𝑤₀^′′, 𝑤₀^′′′.)

4. Recurrence and Transience of the Markov Chains (𝑋_𝑛)_𝑛≥0 and (𝑋^′_𝑛)_𝑛≥0

We have that for the Markov chain (𝑋_𝑛)_𝑛≥0, there is a unique invariant measure up to a multiplicative constant factor 𝜇_𝑘=𝑤_𝑘−1+𝑤_𝑘+𝑤^′_𝑘,   𝑘≥1,𝜇₀=𝑤₀+𝑤^′₀, while for the Markov chain (𝑋^′_𝑛)_𝑛≥0, 𝜇^′_𝑘=𝑤^′′_𝑘−1+𝑤_𝑘^′′+𝑤_𝑘^′′′, 𝑘≥1 with 𝜇^′₀=𝑤₀^′′+𝑤₀^′′′. In the case that an irreducible chain is recurrent there is only one invariant measure (finite or not), so we may obtain the following.

Proposition 3. (i) The Markov chain (𝑋_𝑛)_𝑛≥0 defined as above is positive recurrent if and only if ^∞_𝑘=1𝑏₁⋅𝑏₂⋯𝑏_𝑘1<+∞𝑜𝑟𝑤₀⋅^∞_𝑘=1𝑤_𝑘<+∞_∞𝑘=1,𝛾₁⋅𝛾₂⋯𝛾_𝜅1<+∞𝑜𝑟𝑤^′₀⋅^∞_𝑘=1𝑤^′_𝑘<+∞. (24)
(ii) The Markov chain (𝑋^′_𝑛)_𝑛≥0 defined as above is positive recurrent if and only if ^∞_𝑘=11𝑠₁⋅𝑠₂⋯𝑠_𝑘1<+∞𝑜𝑟𝑤₀^′′⋅^∞_𝑘=1𝑤_𝑘^′′<+∞_∞𝑘=1,𝑡₁⋅𝑡₂⋯𝑡_𝑘1=+∞𝑜𝑟𝑤₀^′′′⋅^∞_𝑘=1𝑤_𝑘^′′′=+∞. (25)

In order to obtain recurrence and transience criterions for the Markov chains (𝑋_𝑛)_𝑛≥0 and (𝑋^′_𝑛)_𝑛≥0 we shall need the following proposition (Karlin and Taylor [6]).

Proposition 4. Let us consider a Markov chain on ℵ which is irreducible. Then if there exists a strictly increasing function that is harmonic on the complement of a finite interval and that is bounded, then the chain is transient. In the case that there exists such a function which is unbounded then the chain is recurrent.

Following this direction, we shall use a well-known method-theorem based on the Foster-Kendall theorem (Karlin and Taylor [6]) by considering the harmonic function 𝑔=(𝑔_𝑘,𝑘≥1). For the Markov chain (𝑋_𝑛)_𝑛≥0 this is a solution of 𝑝₀⋅𝑔₁+𝑟₀⋅𝑔₀=𝑔₀,𝑝_𝑘⋅𝑔_𝑘+1+𝑞_𝑘⋅𝑔_𝑘−1+𝑟_𝑘⋅𝑔_𝑘=𝑔_𝑘,𝑘≥1. (26)

Since Δ𝑔_𝑘=𝑔_𝑘−𝑔_𝑘−1, for every 𝑘≥1, we obtain that 𝑝_𝑘⋅𝑔_𝑘+1+𝑞_𝑘⋅𝑔_𝑘−𝑞_𝑘⋅𝑔_𝑘+𝑞_𝑘⋅𝑔_𝑘−1+𝑟_𝑘⋅𝑔_𝑘=𝑔_𝑘,𝑝_𝑘⋅Δ𝑔_𝑘+1+𝑔_𝑘+𝑞_𝑘⋅𝑔_𝑘−𝑞_𝑘⋅𝑔_𝑘+𝑞_𝑘⋅𝑔_𝑘−1+𝑟_𝑘⋅𝑔_𝑘=𝑔_𝑘𝑝_𝑘⋅Δ𝑔_𝑘+1+𝑝_𝑘+𝑞_𝑘+𝑟_𝑘⋅𝑔_𝑘−𝑞_𝑘⋅𝑔_𝑘+𝑞_𝑘⋅𝑔_𝑘−1=𝑔_𝑘𝑝_𝑘⋅Δ𝑔_𝑘+1−𝑞_𝑘⋅𝑔_𝑘−𝑔_𝑘−1𝑝=0_𝑘⋅Δ𝑔_𝑘+1=𝑞_𝑘⋅Δ𝑔_𝑘. (27)

If we put 𝑚_𝑘=Δ𝑔_𝑘/Δ𝑔_𝑘+1, we get 𝑚_𝑘=𝑝_𝑘/𝑞_𝑘 (with 𝑝_𝑘=1−𝑞_𝑘−𝑟_𝑘), 𝑘≥1, which is the equation of definition of the sequences (𝑠_𝑘)_𝑘≥1 and (𝑡_𝑘)_𝑘≥1 (as a multiplicative factor of the (𝑠_𝑘)_𝑘≥1) for the Markov chain (𝑋^′_𝑛)_𝑛≥0 such that 𝑞^′_𝑘=𝑞_𝑘, 𝑟^′_𝑘=𝑟_𝑘, for every 𝑘≥1. This means that the strictly increasing harmonic functions of the Markov chain (𝑋_𝑛)_𝑛≥0 are in correspondence with the weight representations of the Markov chain (𝑋^′_𝑛)_𝑛≥0 such that 𝑞^′_𝑘=𝑃𝑋^′_𝑛+1=𝑘+1/𝑋^′_𝑛𝑋=𝑘=𝑃_𝑛+1=𝑘−1/𝑋_𝑛=𝑘=𝑞_𝑘,𝑟^′_𝑘=𝑃𝑋^′_𝑛+1=𝑘/𝑋^′_𝑛𝑋=𝑘=𝑃_𝑛+1=𝑘/𝑋_𝑛=𝑘=𝑟_𝑘,𝑝^′_𝑘=1−𝑞^′_𝑘−𝑟^′_𝑘=1−𝑞_𝑘−𝑟_𝑘=𝑝_𝑘,forevery𝑘≥1. (28)

To express this kind of duality we will call the Markov chain (𝑋^′_𝑛)_𝑛≥0, the adjoint of the Markov chain (𝑋_𝑛)_𝑛≥0 and reciprocally in the case that the relation (28) is satisfied.

Equivalently for the Markov chain (𝑋^′_𝑛)_𝑛≥0, the harmonic function 𝑔^′=(𝑔^′_𝑘, 𝑘≥1) satisfies the equation 𝑟^′₀⋅𝑔^′₀+𝑞^′₀⋅𝑔^′₁=𝑔^′₀,𝑞^′_𝑘⋅𝑔^′_𝑘+1+𝑝^′_𝑘⋅𝑔^′_𝑘−1+𝑟^′_𝑘⋅𝑔^′_𝑘=𝑔^′_𝑘,𝑘≥1. (29)

Since Δ𝑔^′_𝑘=𝑔^′_𝑘−𝑔^′_𝑘−1, for every 𝑘≥1, we have 𝑞^′_𝑘⋅Δ𝑔^′_𝑘+1+𝑔^′_𝑘+𝑝^′_𝑘⋅𝑔^′_𝑘−𝑝^′_𝑘⋅𝑔^′_𝑘+𝑝^′_𝑘⋅𝑔^′_𝑘−1+𝑟^′_𝑘⋅𝑔^′_𝑘=𝑔^′_𝑘,𝑝^′_𝑘+𝑞^′_𝑘+𝑟^′_𝑘⋅𝑔^′_𝑘+𝑞^′_𝑘⋅Δ𝑔^′_𝑘+1−𝑝^′_𝑘⋅𝑔^′_𝑘+𝑝^′_𝑘⋅𝑔^′_𝑘−1=𝑔^′_𝑘,𝑞^′_𝑘⋅Δ𝑔^′_𝑘+1=𝑝^′_𝑘⋅𝑔^′_𝑘−𝑔^′_𝑘−1=𝑝^′_𝑘⋅Δ𝑔^′_𝑘. (30)

If we put ℓ_𝑘=Δ𝑔^′_𝑘+1/Δ𝑔^′_𝑘, we get ℓ_𝑘=𝑝^′_𝑘/𝑞^′_𝑘 (with 𝑞^′_𝑘=1−𝑝^′_𝑘−𝑟^′_𝑘), 𝑘≥1, which is the equation of the definition of the sequences (𝑏_𝑘)_𝑘≥1 and (𝛾_𝑘)_𝑘≥1 (as a multiplicative factor of the (𝑏_𝑘)_𝑘≥1) for the Markov chain (𝑋_𝑛)_𝑛≥0 such that 𝑝^′_𝑘=𝑝_𝑘, 𝑟^′_𝑘=𝑟_𝑘, for every 𝑘≥1. By considering a similar approximation of that given before for the Markov chain (𝑋_𝑛)_𝑛≥0, we may say that the strictly increasing harmonic functions of the Markov chain (𝑋^′_𝑛)_𝑛≥0 are in correspondence with the weight representations of the Markov chain (𝑋_𝑛)_𝑛≥0 such that equivalent equations of (28) are satisfied.

So we may have the following.

Proposition 5. The Markov chain (𝑋_𝑛)_𝑛≥0 defined as above is transient if and only if the adjoint Markov chain (𝑋^′_𝑛)_𝑛≥0 is positive recurrent and reciprocally. Moreover the adjoint Markov chains (𝑋_𝑛)_𝑛≥0 and (𝑋^′_𝑛)_𝑛≥0 are null recurrent simultaneously.
In particular (i)the Markov chain (𝑋_𝑛)_𝑛≥0 defined as above is transient if and only if (1/𝑤₀^′′)⋅∑^∞_𝑘=1𝑤_𝑘^′′<+∞ and (1/𝑤₀^′′′)⋅∑^∞_𝑘=1𝑤_𝑘^′′′=+∞;(ii)the Markov chain (𝑋^′_𝑛)_𝑛≥0 defined as above is transient if and only if (1/𝑤₀)⋅∑^∞_𝑘=1𝑤_𝑘<+∞ and (1/𝑤^′₀)⋅∑^∞_𝑘=1𝑤^′_𝑘<+∞;(iii)the adjoint Markov chains (𝑋_𝑛)_𝑛≥0 and (𝑋^′_𝑛)_𝑛≥0 are null recurrent if1𝑤₀⋅^∞_𝑘=1𝑤_𝑘=1𝑤₀^′′⋅^∞_𝑘=1𝑤_𝑘^′′1=+∞,𝑤^′₀⋅^∞_𝑘=1𝑤^′_𝑘=1𝑤₀^′′′⋅^∞_𝑘=1𝑤_𝑘^′′′=+∞. (31)

Proof. The proof of Proposition 5 is an application mainly of Proposition 4 as well as of Proposition 3.

Acknowledgment

C. Ganatsiou is indebted to the referee for the valuable comments, which led to a significant change of the first version.