Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically
William Oçafrain
math.PR











[1]Institut de Mathématiques de Toulouse. CNRS UMR 5219. Université Paul Sabatier, 118 route de Narbonne, F-31062 Toulouse cedex 09.

  E-mail: william.ocafrain@math.univ-toulouse.fr

Quasi-stationarity and quasi-ergodicity for discrete-time Markov chains with absorbing boundaries moving periodically
    William Oçafrain^1
    December 30, 2023
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    We are interested in quasi-stationarity and quasi-ergodicity when the absorbing boundary is moving. First we show that, in the moving boundary case, the quasi-stationary distribution and the quasi-limiting distribution are not well-defined when the boundary is oscillating periodically. Then we show the existence of a quasi-ergodic distribution for any discrete-time irreducible Markov chain defined on a finite state space in the fixed boundary case. Finally we use this last result to show the quasi-ergodicity in the moving boundary case.   


 Key words :  Quasi-stationary distribution, quasi-limiting distribution, quasi-ergodic distribution, Q-process, periodic moving boundaries, random walk


 2010 Mathematics Subject Classification: 60B10; 60F99 47D03 , 60J10 




§ INTRODUCTION


Let (Ω,  A, ) be a probability space and let X = (X_n)_n ∈ be a Markov chain with a finite state space (E,  E), E being the σ-algebra containing all the subset of E. Let _x be the probability measure such that _x(X_0 = x)=1 and, for any measure μ on E, define _μ = ∫_x dμ(x). Denote by _1(E) the set of probability measures defined on E. 

We define, for each time n ≥ 0, a subset A_n ⊂ E called killing subset at time n and we denote by E_n the complement of A_n called survival subset at time n. We will call (A_n)_n ∈ the moving killing subset or the moving killing boundary. 
We denote by τ the random variable defined as follows

    τ := inf{n ≥ 0 : X_n ∈ A_n}

For any subset B ⊂ E, we define τ_B as follows

    τ_B := inf{n ≥ 0 : X_n ∈ B}

and, to make the notation easier, for any m ∈, we denote by τ_m the random variable defined by

    τ_m := τ_A_m = inf{n ≥ 0 : X_n ∈ A_m}

 This chapter will deal with quasi-stationary, quasi-limiting and quasi-ergodic distributions that we define as follows. 


ν is a quasi-stationary distribution if for any n ≥ 0

    _ν(X_n ∈· | τ > n) = ν(·)





ν is a quasi-limiting distribution if there exist some μ∈ M_1(E) such that

    _μ(X_n ∈· | τ > n) n →∞⟶ν(·)





ν is a quasi-ergodic distribution or a mean-ratio quasi-stationary distribution if for any μ∈ M_1(E) and any bounded measurable function f

    _μ(1/nk=0n-1∑ f(X_k)| τ > n) n →∞⟶∫ f dν


We will also be interested in the existence of a Q-process, which can be interpreted as the process X conditioned never to be absorbed by (A_n)_n ∈. 
 

In the case where the sequence (A_n)_n ∈ does not depend on the time, the existence of these probability measures was established under several assumptions. See for example <cit.> for a general review on the theory of quasi-stationary distributions.  
For modelling purpose, some recent works (see <cit.>) introduce some Markov processes absorbed by moving boundaries and the classical theory on quasi-stationary distributions does not allow anymore to describe the asymptotic behavior of the process conditioned not to be absorbed. Our purpose is therefore to check whether these probability measures are still well-defined when (A_n)_n ≥ 0 depends on the time or not. 


In all what follows, we will assume that for any x ∈ E_0,

    _x(τ < ∞) = 1

and will also assume that the following hypothesis of irreducibility holds

    ∀ n ∈, ∀ x,y ∈ E_n, ∃ m ∈, _x(X_m τ_n=y) > 0


Quasi-stationary distribution will be studied for general moving killing boundaries. However, in a significant part of this chapter we will deal with moving killing boundaries (A_n)_n ∈ which are γ-periodic with γ≥ 2. 

In this chapter, we will actually show that there are no quasi-stationary distributions and quasi-limiting distributions in the sense of Definitions <ref> and <ref> when the boundaries are moving periodically. However, we will show that the notion of quasi-ergodic distribution and Q-process still makes sense even when the boundary is moving. In particular, we will show the following statement.

Assume that (A_n)_n ∈ is γ-periodic. Then, for some initial law μ∈ P(E) and under assumptions which will be spelled out later, there exists a probability measure η such that, for any bounded measurable function f,

    _μ(1/nk=0n-1∑ f(X_k) | τ > n) n →∞⟶∫ f dη


The proof is divided to several steps. First we reduce the problem to the study of quasi-stationary distribution in a non moving domain, but for a periodic Markov chain. Then we extend the result proved by Darroch and Senata (see <cit.>) in the aperiodic case to the periodic situation (γ∈^*). 

This chapter ends with an application of this theorem to random walks absorbed by 2-periodic moving boundaries.



§ QUASI-STATIONARY DISTRIBUTION WITH MOVING KILLING SUBSET


The following proposition shows that the notion of quasi-stationary distribution as defined in Definition <ref> is not relevant when the killing boundary is moving.


Assume there exist l,m ∈ such that A_l  A_m. Then there is no measure ν∈ M_1(E) satisfying the following property:

    ∀ n ∈,   _ν(X_n ∈· | τ > n) = ν(·)



 For any n ∈, denote by f_n :  M_1(E) → M_1(E) the function defined by

    f_n : μ⟶ℙ_μ(X_1 ∈· | τ_n > 1)

where τ_n is defined in (<ref>) and denote by μ_n the probability measure defined by

    μ_n = _μ(X_n ∈· | τ > n)

By the Markov property, we have for any n ∈^*

    μ_n = _μ_n-1(X_1 ∈· | τ_n > 1) = f_n(μ_n-1)

Thus, by induction, we obtain for any n ∈

    ℙ_μ(X_n∈· | τ > n) = f_n∘…∘ f_1(μ)

We deduce from this equality that

    ∀ n ∈, _ν(X_n∈· | τ > n) = ν(·)     ⟺∀ n ∈,  f_n(ν) = ν
       ⟺∀ n ∈, _ν(X_1 ∈· | τ_n > 1) = ν(·)

In other words, ν is a quasi-stationary distribution in the moving sense if and only if it is a quasi-stationary distribution in the non-moving sense for all the subsets A_n.
In particular, if ν satisfies (<ref>),

    ν(·) = _ν(X_1 ∈· | τ_l > 1)  and ν(·) = _ν(X_1 ∈· | τ_m > 1)

where l and m have been mentioned in the statement of the proposition. However, since the assumption of irreducibility (<ref>) holds, the previous statement is impossible since the support of the quasi-stationary distributions are different. 


Proposition <ref> can be extended to any general Markov process defined on any space state. In particular, for continuous-time Markov processes defined on a metric space (E,d), we may replace the assumption of irreducibility (<ref>) by the following assumption

    ∀ t ∈_+, ∀ x,y ∈ E_t, ∀ϵ > 0, ∃ t_0 ∈_+, _x(X_t_0 τ_t∈ B(y,ϵ)) > 0

where B(y,ϵ) := {z ∈ E : d(y,z) < ϵ}.


Notice moreover that we did not need any assumption about the behavior of (A_n)_n ∈. In all what follows, we consider that (A_n)_n ∈ is γ-periodic with γ≥ 2.



§ QUASI-LIMITING DISTRIBUTION WHEN THE KILLING SUBSET IS MOVING PERIODICALLY


We are now interested in knowing whether the definition of quasi-limiting distribution given in definition <ref> is meaningful when the killing subset is moving or not. 
In the usual case, it is well known (see <cit.> p. 345) that quasi-stationary distribution and quasi-limiting distribution are equivalent notions. This implies that the non-existence of a quasi-stationary distribution implies the non-existence of any quasi-limiting distribution. However, this equivalence does not hold anymore in the moving case. Consider for example  (A_n)_n ≥ 0 such that there exists n_0 such that for any n ≥ n_0, A_n = A_n_0
and assume that there exists a quasi-stationary distribution ν_n_0 (in the non-moving sense) such that for any probability measure μ on E_n_0,

    _μ(X_n ∈· | τ_n_0 > n) n →∞⟶ν_n_0

Thus, by the Markov property, for any μ∈ M_1(E) such that _μ(τ > k) > 0 for all k ∈ and any n ≥ 0,

    _μ(X_n+n_0∈· | τ > n+n_0) = _μ_n_0(X_n ∈· | τ_A_n_0 > n) n →∞⟶ν_n_0

where μ_n is defined in (<ref>) for any n ∈.
 From now on, we will assume that (A_n)_n ∈ is periodic and will denote by γ its period. We will show that quasi-limiting distribution is not well defined when the killing subset is moving periodically.


Assume (A_n)_n ∈ is γ-periodic and there exist 0 ≤ l,m ≤γ -1 such that A_l  A_m. 

Then there is no ν∈ M_1(E) satisfying the following property:

    ∃μ∈ M_1(E),   _μ(X_n ∈· | τ > n) n →∞⟶ν(·)




Consider again the functions f_m defined in (<ref>):

    f_m : μ⟶_μ(X_1 ∈· | τ_m > 1)

Then using the periodicity of (A_n)_n ∈ and by the Markov property, for any k ∈{1, …, γ}, m ∈ℕ^* and μ∈ M_1(E)

    ℙ_μ(X_k + m γ∈· | τ > k+m γ) = g_k ∘ f^m(μ)

with 

    g_k = f_k∘…∘ f_1
       f = f_γ∘…∘ f_1

 Assume that there exists μ∈ M_1(E) such that the sequence (ℙ_μ(X_m∈· | τ > m))_m ∈ℕ converges to its limit ν. Then

    ν   = m →∞limℙ_μ(X_mγ∈· | τ > mγ) 
       = m →∞lim f^m(μ)

So for any k ∈{1, …, γ}

    ν = g_k(ν) = ℙ_ν(X_k∈· | τ > k)

In other words, for any k ∈{1, …, γ},

    ν = f_k(ν)

We thus may conclude in the proof of proposition <ref>.


The previous statement implies therefore that the quasi-limiting distribution as defined in Definition <ref> is not well-defined when the moving killing subset is periodic. However, according to the proof of the previous proposition, it seems that the sequence of these conditioned probabilities could have some limit points. 


The following proposition allows us to pass from a moving problem to a non-moving problem. The existence of limit points will be therefore a consequence of the existence of classical quasi-stationary distributions for the transformed Markov chain. 

For any 0 ≤ m ≤γ-1 and μ∈ M_1(E), there is a Markov chain (X^(m)_n)_n ∈ℕ such that 

    ℙ_μ((X_m, X_m+γ, …, X_m + nγ) ∈· | τ > m + nγ) = ℙ_μ_m((X^(m)_0, X^(m)_1, …, X^(m)_n) ∈· | τ^(m)_m > n)

where μ_m is defined in (<ref>) and τ^(m)_m = inf{n ∈ : X^(m)_n ∈ A_m}. 


According to the Markov property, it is enough to show that for any γ-periodic sequence of subsets B = (B_n)_n ∈ and any measure μ∈ M_1(E), there exists a Markov chain (Z_n)_n ∈ℕ such that

    ℙ_μ((X_γ, …, X_nγ) ∈· | τ(B) >  nγ) = ℙ_μ((Z_1, …, Z_n) ∈· | τ̃_B_0 > n)

where τ(B) = inf{m ≥ 0 : X_m ∈ B_m} and τ̃_B_0 = inf{n ∈ : Z_n ∈ B_0}. Denote F_0 the complement of B_0. 
For any x ∈ F_0 define p(x,·) by

    p(x,A) = ℙ_x(X_γ∈ A, τ_B > γ),   ∀ A ⊂ F_0 
        p(x,B_0) = 1-p(x,F_0)
 and we denote by (Z_n)_n ∈ℕ the Markov chain for which the transition kernel is p. We will show by induction that, for any ϕ_1, …, ϕ_n bounded measurable functions, 

    𝔼_μ(ϕ_1(X_γ) …ϕ_n(X_nγ) 1_τ(B) > nγ) = 𝔼_μ(ϕ_1(Z_1) …ϕ_n(Z_n) 1_τ̃_B_0 > n)

By definition of (Z_n)_n ∈, for any probability measure μ and any bounded measurable function ϕ,

    _μ(ϕ(Z_1)_τ̃_B_0 > 1) = _μ(ϕ(X_γ) _τ(B) > γ)

which entails the base case.
Now assume that the equality for n-1 is satisfied. Let ϕ_1, …, ϕ_n be some bounded measurable functions. Then

    𝔼_μ(ϕ_1(X_γ) …ϕ_n(X_nγ) 1_τ(B) > nγ)    = 𝔼_μ(ϕ_1(X_γ) 1_τ(B) > γ𝔼_X_γ(ϕ_2(X_γ) …ϕ_n(X_(n-1)γ) 1_τ(B) > (n-1)γ)) 
       =𝔼_μ(ϕ_1(Z_1) 1_τ̃_B_0 > 1𝔼_Z_1(f_2(Z_1) …ϕ_n(Z_n-1) 1_τ̃_B_0 > (n-1))) 
       = 𝔼_μ(ϕ_1(Z_1) …ϕ_n(Z_n) 1_τ̃_B_0 > n)

This concludes the proof.




§ EXISTENCE OF QUASI-ERGODIC DISTRIBUTION WITH PERIODIC MOVING KILLING SUBSETS

In this section, our aim is to show the existence of a quasi-ergodic distribution as defined in Definition <ref> when the boundary is moving periodically. This section will be split into three parts : 


  * We will first study quasi-ergodicity in the non-moving case (when A_n = A_0, ∀ n ∈) for irreducible Markov chains.

  * Then we will use the results obtained in the first part to deduce quasi-ergodicity for general Markov chains (irreducible or not), but still considering non-moving boundaries

  * Finally we will show the existence of quasi-ergodic distribution when (A_n)_n ∈ is moving periodically.  
 


 §.§ Quasi-ergodic distribution in the classical non-moving sense for irreducible periodic Markov chains


In this subsection we will study the quasi-ergodicity of one irreducible Markov chain Y = (Y_n)_n ∈ in the classical non-moving sense, that is when the killing edge does not move. Without loss of generality, assume Y is defined in the state space E_0 ={0, …, K} and that the cemetery is {0}. In this subsection and the following, τ will be defined as (<ref>) but refering to Y, that is

    τ = inf{n ≥ 0 : Y_n = 0}

 Denote by P the transition matrix of Y. Since 0 is an absorbing state for Y, P has the following form

    P = [ 1 0; v Q ]

where Q is the sub-transition matrix. We will assume that Q is irreducible (i.e. ∀ x,y ∈ E_0, ∃ n ∈, Q^n(x,y) > 0). As a result we can define T_x the period of x as

    T_x := gcd{n ∈ :  _x(Y_n = x, τ > n) > 0}

where gcd refers to the greatest common divisor. By irreducibility of Q, all the x have the same period and we denote by T this common period.

The existence of quasi-ergodic distributions has already been proved by Darroch and Seneta in <cit.> when T=1. However we will see that this result is not enough for our purpose and we need to extend it to periodic Markov chains.

Due to the periodicity of Q, there exist (C_i)_0 ≤ i ≤ T-1 a partition of E_0 such that if the support of the initial distribution μ is included in C_0, then for any n ∈ and 0 ≤ k ≤ T-1,

    _μ(Y_k+nT∈ C_k, τ > k+nT) = 1
 
Without loss of generality, we construct (C_i)_0 ≤ i ≤ T-1 such that 1 ∈ C_0. Formally (C_i)_0 ≤ i ≤ T-1 are defined by

    C_0 := {y ∈ E_0 : ∃ n ∈^*, _1(Y_nT = y,τ > nT)> 0}
       ∀ 1 ≤ i ≤ T-1,  C_i := {y ∈ E_0 : ∃ x ∈ C_i-1, _x(Y_1 = y)>0}

For each j ∈{0, …, T-1} and any v ∈^K, we will denote by v^(j) the sub-vector of v restricted to C_j. 

It is well known by the Perron-Frobenius theorem that the spectral radius 
    ρ := max{|λ| : λ∈ Sp(Q)}
 is a simple eigenvalue of Q and that one can find a left-eigenvector ν = (ν(j))_1 ≤ j ≤ K and a right-eigenvector ξ = (ξ(j))_1 ≤ j ≤ K for ρ (i.e. ν Q = ρν and Qξ = ρξ) such that ν(j) > 0 and ξ(j) > 0 for all j ∈{1, …, K}. We may choose ν and ξ such that

    <ν,> = <ν,ξ> = 1

 where <·,·> is the usual Hermitian product on ^K. Moreover, since Q is T-periodic, 

    {λ_k := ρ e^2ikπ/T : 0 ≤ k ≤ T-1}⊂ Sp(Q)

and each λ_k is simple. For each λ_k we can obtain a left eigenvector v_k and a right-eigenvector w_k from ν and ξ with the following transformation

    ∀ j ∈{0, …, T-1}, v_k^(j) = e^-i2π j k/Tν^(j) and  w_k^(j) = e^i2π j k/Tξ^(j)

See Theorem 1.7 in [<cit.>,p.23-24] for more details. 

The vectors (v_i)_0 ≤ i ≤ T-1 are linearly independent. We can complete this family into a basis V = (v_i)_0 ≤ i ≤ K-1 such that v_i ∈ Span^⊥(v_0, …, v_T-1) for all T ≤ i ≤ K-1 where

    Span^⊥(v_0, …, v_T-1) = {v ∈^K : <v,v_i> = 0,  ∀ i ∈{0, …, T-1}}

 Let us denote by R the matrix representing the change of basis from the canonical basis to V. Then we have the following decomposition

    Q = R [ λ_0       
       ⋱   
          λ_T-1                              0;                              0                            Q_0 ]
        R^-1

where Q_0 is a (K-T) × (K-T) matrix. We define the matrix Q' by

    Q' = R [   0   0;   0 Q_0 ] R^-1




Let f : {1, …, K}→ be a bounded measurable function. Then for any x ∈{1, …, K} and n ∈^*,  

    _x(1/nk=0n-1∑ f(Y_k) _τ > n) = ρ^n φ(f) l=0T-1∑ e^-2inlπ/T  <w_l,δ_x> <v_l,> + o(ρ^n)

where 
    φ(f) = i=1K∑ f(i)ν(i) ξ(i)
.


Let f:{1, …, K}→ be a bounded measurable function. In this proof we will consider probability measures on {1, …, K} and functions defined on {1, …, K} as K-vectors. Thus for any x ∈{1, …, K} we can say

    _x(f(Y_n)_τ > n) = <δ_x Q^n, f>

where δ_x is the Dirac measure on x.
For any x ∈{1, …, K}, define (α_k(x))_0 ≤ k ≤ T-1 the unique family of ^K such that there is μ_x ∈ Span^⊥(v_0, …, v_T-1) such that 

    δ_x = k=0T-1∑α_k(x) v_k + μ_x

We will use the following lemma whose proof is postponed after the proof of the theorem.


For any 0 ≤ k ≤ T-1,

    (α_k(x))_x ∈ E_0 = w_k

where w_k is defined in (<ref>) 

Thus we can write

    δ_x = k=0T-1∑ w_k(x) v_k + μ_x

So, using (<ref>) and (<ref>), for any n ∈

    𝔼_x(f(X_n)1_τ > n)
       = <k=0T-1∑ w_k(x) v_k Q^n, f> + <μ_x Q^n,f> 
       = k=0T-1∑λ_k^n w_k(x) <v_k, f> + <μ_x(Q')^n,f>
 
Now, using the Markov property, for any k ≤ n, 

    𝔼_μ(f(Y_k)1_τ > n)    = 𝔼_μ(1_τ > k f(Y_k) ℙ_Y_k(τ > n-k)) 
       =𝔼_μ(1_τ > kg_k,n(f)(Y_k))

where, for any y ∈ E_0,  
    g_k,n(f)(y) = f(y)_y(τ > n-k)
 Then,

    g_k,n(f)(y)
           = f(y) <δ_y Q^n-k, 1> 
       = m=0T-1∑λ_m^n-k f(y) w_m(y)<v_m,1> + f(y)<μ_y (Q')^n-k,>

Define, for any k ∈{0, …, T-1} and n ∈,

    g_k(f):y → f(y)w_l(y)
       w_n(f) : y → f(y) <μ_y(Q')^n,>
Then, using (<ref>), for any k ≤ n,

    𝔼_x(f(Y_k)1_τ > n)    =<δ_x Q^k, g_k,n(f)>
       = l=0T-1∑λ_l^k w_l(x) <v_l, g_k,n(f)> + <μ_x(Q')^k,g_k,n(f)> 
       = A_k,n+B_k,n+C_k,n+D_k,n

where

    A_k,n = l=0T-1∑m=0T-1∑λ_l^k λ_m^n-kw_l(x)<v_l,g_m(f)><v_m,> 
       B_k,n = l=0T-1∑λ_l^k w_l(x) <v_l, w_n-k(f)> 
       C_k,n = m=0T-1∑λ_m^n-k <v_m,1> <μ_x(Q')^k,g_m(f)> 
       D_k,n = <μ_x(Q')^k,w_n-k(f)>

Hence for any n ∈^*

    k=0n-1∑𝔼_x(f(Y_k)1_τ > n) = k=0n-1∑ A_k,n +k=0n-1∑ B_k,n +k=0n-1∑ C_k,n +k=0n-1∑ D_k,n

i) Study of k=0n-1∑ A_k,n

For any n ∈^*,

    k=0n-1∑ A_k,n   =  l=0T-1∑m=0T-1∑(k=0n-1∑λ_l^k λ_m^n-k)w_l(x)<v_l,g_m(f)><v_m,> 
       = l=0T-1∑ n λ_l^n w_l(x) <v_l,g_l(f)><v_l,> + l  m∑λ_m(λ_l^n - λ_m^n/λ_l-λ_m) w_l(x)<v_l,g_m(f)><v_m,>

On the one side,

    l=0T-1∑ n λ_l^n w_l(x) <v_l,g_l(f)><v_l,> = nρ^n l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,g_l(f)><v_l,>

On the other side, for any 0 ≤ l  m ≤ T-1,

    λ_m(λ_l^n - λ_m^n/λ_l-λ_m)    = ρ  e^-2imπ/T(ρ^n  e^-2inlπ/T -ρ^n  e^-2inmπ/T/ρ  e^-2ilπ/T-ρ  e^-2imπ/T) 
       = ρ^n  e^-2imπ/T(  e^-2inlπ/T -  e^-2inmπ/T/  e^-2ilπ/T-  e^-2imπ/T)

(e^-2imπ/T(  e^-2inlπ/T -  e^-2inmπ/T/  e^-2ilπ/T-  e^-2imπ/T))_n ∈ is bounded, hence 

    1/n× e^-2imπ/T(  e^-2inlπ/T -  e^-2inmπ/T/  e^-2ilπ/T-  e^-2imπ/T) n →∞⟶ 0

We deduce that, for any 0 ≤ l  m ≤ T-1,

    ρ^n  e^-2imπ/T(  e^-2inlπ/T -  e^-2inmπ/T/  e^-2ilπ/T-  e^-2imπ/T) = o(nρ^n)

and therefore

    l  m∑λ_m(λ_l^n - λ_m^n/λ_l-λ_m) w_l(x)<v_l,g_m(f)><v_m,> = o(nρ^n)

since this is a finite sum. Hence

    k=0n-1∑ A_k,n =  nρ^n l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,g_l(f)><v_l,> + o(nρ^n)

ii) Study of k=0n-1∑ B_k,n

For any y ∈ E, n ∈ and 0 ≤ l ≤ T-1

    k=1n-1∑λ_l^k w_n-k(f)(y)    = f(y) <μ_y (k=0n-1∑λ_l^k (Q')^n-k),> 
       = f(y) <μ_yQ'(λ_l I_K - Q')^-1(λ_l^n I_K-(Q')^n),>

where I_K is the K × K-identity matrix. For any 0 ≤ l ≤ T-1 and n ∈,

    λ_l^n I_k - (Q')^n = ρ^n (e^2iπ nl/T I_k - ρ^-n(Q')^n)

and (e^2iπ nl/T I_k - ρ^-n(Q')^n)_n ∈ is bounded since the spectral radius of Q' is smaller than ρ. Hence 

    1/n(e^2iπ nl/T I_k - ρ^-n(Q')^n) n →∞⟶ 0

where 0 is understood as the zero matrix, and we deduce that 

    <μ_yQ'(λ_l I_K - Q')^-1(λ_l^n I_K-(Q')^n),> = o(nρ^n)

As a result, for any n ∈,

    k=1n-1∑λ_l^k w_n-k(f)(y) = o(nρ^n)

Hence for any n ∈

    k=0n-1∑ B_k,n = l=0T-1∑w_l(x) <v_l,k=0n-1∑λ_l^k w_n-k(f)> = o(nρ^n)

iii) Study of k=0n-1∑ C_k,n

In the same way as k=0n-1∑ B_k,n,

    k=0n-1∑ C_k,n   = ∑_k=0^n-1∑_m=0^T-1λ_m^n-k <v_m, > <μ_x (Q')^k, g_m(f)> 
       = ∑_m=0^T-1 <v_m,> <μ_x ( ∑_k=0^n-1λ_m^n-k (Q')^k), g_m(f)>

For any 0 ≤ m ≤ T-1 and n ≥ 1,

    ∑_k=0^n-1λ_m^n-k (Q')^k = λ_m × (λ_m I_K - Q')^-1 (λ_m^n I_K-(Q')^n)

We already showed that for any 0 ≤ m ≤ T-1 and n ≥ 1
 
    (λ_m I_K - Q')^-1 (λ_m^n I_K-(Q')^n) = o(n ρ^n)

Finally,

    k=0n-1∑ C_k,n = o(nρ^n)

iv) Study of k=0n-1∑ D_k,n

Finally, let us denote by (q')_i,j^(n), for i,j ∈{1, …, K-T} and n ∈, the coefficient of (Q')^n located at the ith row and the jth column. Then for any n ∈

    k=0n-1∑ D_k,n = i,j,l,m∑μ_x(j) f(i) μ_i(m) (k=0n-1∑(q')_m,l^(n-k)(q')_i,j^(k))

Let i,j,l,m ∈{1, …, K}. By definition of the matrix Q', the spectral radius of Q' is strictly smaller than ρ. We deduce from this 

    (q')^(n)_i,j = o(ρ^n),   (q')^(n)_m,l = o(ρ^n)

In particular there is a positive number C such that for any n ∈ and m,l ∈{1, …, K},

    (q')^(n-k)_m,l≤ C ρ^n-k

Hence,

    k=0n-1∑(q')_m,l^(n-k)(q')_i,j^(k)   ≤ Ck=0n-1∑ρ^n-k(q')_i,j^(k)
       = C n ρ^n (1/nk=1n-1∑ρ^-k(q')_i,j^(k))

However, by (<ref>), ρ^-nq_i,j^(n)→ 0 when n tends to infinity and using Cesaro's lemma,

    1/nk=0n-1∑ρ^-k(q')_i,j^(k)n →∞⟶ 0

Thus using (<ref>) and (<ref>), we deduce that

    k=0n-1∑ D_k,n = o(nρ^n)

Hence, gathering all these results and using (<ref>),

    k=0n-1∑𝔼_x(f(Y_k)1_τ > n) = nρ^n l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,g_l(f)><v_l,> + o(nρ^n)

However we have for any l ∈{0, …, T-1}

    <v_l, g_l(f)>    = j=1K∑ f(j) v_l(j) w_l(j)
       = j=0T-1∑x ∈ C_j∑ f(x) v_l(x) w_l(x)
       = j=0T-1∑x ∈ C_j∑ f(x) e^-i2π j l/Tν(x) e^i2π j l/Tξ(x)
       = <v_0, g_0(f)>

As a result,

    _x(k=0n-1∑ f(Y_k) _τ > n) = nρ^n <v_0,g_0(f)> l=0T-1∑ e^-2inlπ/Tw_l(x) <v_l,> + o(nρ^n)


Now we prove Lemma <ref> quoted in the previous proof.

Let us start by proving that α_l is a right-eigenvector associated to λ_l. Since Q is a real matrix, it is equivalent to show that α_l is a right-eigenvector associated to λ_l. 

First remind that α_l is defined by the relations

    δ_k = l=0T-1∑α_l(k) v_l + δ_k'

for any k ∈ E_0 and with δ_k' ∈ Span^⊥(v_0, …, v_T-1). This implies for any k 

    <δ_k, v_m> = l=0T-1∑α_l(k) <v_l,v_m>

or, in other words,

    [   <δ_k, v_0>;            ⋮; <δ_k, v_T-1> ] = [      <v_0,v_0>              …    <v_T-1,v_0>;              ⋮              ⋱              ⋮;   <v_0, v_T-1>              … <v_T-1, v_T-1> ][   α_0(k);        ⋮; α_T-1(k) ]

Denote by A the matrix

    A = [      <v_0,v_0>              …    <v_T-1,v_0>;              ⋮              ⋱              ⋮;   <v_0, v_T-1>              … <v_T-1, v_T-1> ]

A is simply the Gram's matrix of the basis (v_i)_0 ≤ i ≤ T-1. Thus the determinant (A) is positive and for any x ∈ E_0

    α_l(x) = 1/(A)
        <v_0,v_0>    …    <δ_x, v_0>     …    <v_T-1,v_0> 
    ⋮   ⋱   ⋮   ⋱   ⋮
    
        <v_0,v_T-1>    …    <δ_x, v_T-1>    …    <v_T-1,v_T-1>

where the column (<δ_x, v_0>, …, <δ_x,v_T-1>)^T is the l-th columns of the matrix.
We want to show now that α_l is a right-eigenvector associated to λ_l, that is

    ∀ v ∈^K, <v,Qα_l> = λ_l <v,α_l>

In fact it is enough to show (<ref>) when v is one of left-eigenvectors and when v ∈ Span^⊥(v_0, …, v_T-1). 
In the case where v = v_k for k ∈{0, …, T-1}

    <v_k, α_l>    = j=1K∑v_k(j)1/(A)
        <v_0, v_0>    …    <δ_j, v_0>    …     <v_T-1,v_0> 
    ⋮   ⋱   ⋮   ⋱   ⋮
    
        <v_0, v_T-1>    …    <δ_j, v_T-1>    …    <v_T-1,v_T-1>
        
       = 1/(A)
        <v_0, v_0>    …    <v_k, v_0>    …     <v_T-1,v_0> 
    ⋮   ⋱   ⋮   ⋱   ⋮
    
        <v_0, v_T-1>    …    <v_k, v_T-1>    …     <v_T-1,v_T-1>
        
       ={[   1 l=k;   0     ].

We deduce from this 

    <v_k, Q α_l> = λ_l <v_k, α_l>,   ∀ 0 ≤ k ≤ T-1

Finally, for any v ∈ Span(v_0, …, v_T-1)^⊥,

    <v,α_l> = 1/(A)
        <v_0, v_0>    …    0    …     <v_T-1,v_0> 
    ⋮   ⋱   ⋮   ⋱   ⋮
    
        <v_0, v_T-1>    …    0    …     <v_T-1,v_T-1>
        
        =0

Thus we have

    <v,Qα_l> = 0 = λ_l <v_k, α_l>

because ^tQv ∈ Span(v_0, …, v_T-1)^⊥. 

Hence for each k ∈{0, …, T-1}, there is β_k ∈ such that α_k = β_k w_k (where w_k is defined at the beginning of the subsection). We will show that β_k = β_0 = 1 for any k. 

Remark that A can be written as i=1T∑ a_i-1 P_σ_i where P_σ_i is the permutation matrix of σ_i where σ_i = (i  i+1  … i-2  i-1) and a_0 > 0 and a_1, …, a_T-1∈. In other words, A is of the following shape

    A = [   a_0   a_1   a_2     … a_T-1; a_T-1   a_0   a_1     … a_T-2;     ⋮     ⋮     ⋱     ⋮;   a_1   a_2   a_3     …   a_0 ]

with a_0 > 0 and a_1, …, a_T-1∈^T-1. Moreover, since 1 ∈ C_0, <δ_1, v_l> = <δ_1, v_0> = ν_1 for any l ∈{0, …, T-1}. As a result, for any l ∈{0, …, T-1},

    (A) α_l(1)    = 
        a_0    …   ν_1    …     a_T-1
    ⋮   ⋱   ⋮   ⋱   ⋮
    
        a_1    …   ν_1    …     a_1
        
       = ν(1)     a_l+1   …   …     a_l-1
    ⋮   ⋮   ⋱   ⋱   ⋮
    ν(1)     a_l+2   …   …     a_l
        
       = ν(1)     a_1   …   …     a_T-1
    ⋮   ⋮   ⋱   ⋱   ⋮
    ν(1)     a_2   …   …     a_0
        
       =(A) α_0(1)

Indeed, from (<ref>) to (<ref>), we applied a circular permutation for the columns in order to put the vector ^t(ν(1), …, ν(1)) at the first column, and the determinant stays the same after this transformation. From (<ref>) to (<ref>), we did a circular permutation on the rows, which does not affect either the determinant. 

We deduce from this equality that β_k = β_0 for any k ∈{0, …, T-1} because w_k(1) = w_0(1). Concerning the fact that β_0 = 1, remark that 

    i=1K∑ν(i) α_0(i)   = i=1K∑v_0(i) α_0(i)
       = 1/(A)
        <v_0,v_0>    …    <v_T-1,v_0> 
    ⋮   ⋱   ⋮
    
        <v_0, v_T-1>    …    <v_T-1, v_T-1> 
        
       =1

And 

    i=1K∑ν(i) α_0(i) = β_0 i=1K∑ν(i) ξ(i) = 1


The statement of Theorem 1 is meaningful provided the coefficient of the leading term ρ^n is not equal to 0. In the following proposition we prove that this coefficient is actually not 0.  


For any n ∈ and any x 

    l=0T-1∑ e^-2inlπ/T <w_l,δ_x> <v_l,>  0



Let x ∈ E_0. Then there exists k ∈{0, …, T-1} such that x ∈ C_k. Thus, for any n ∈,

    l=0T-1∑ e^-2inlπ/Tw_l(x)  <v_l,> 
        
           = l=0T-1∑ e^-2i(n+k)lπ/Tξ(x)(j=0T-1∑y ∈ C_j∑ e^2iπ l j/Tν(y))  
       =j=0T-1∑y ∈ C_j∑ξ(x) ν(y) ( l=0T-1∑ e^-2iπ (n+k-j)l/T) 
    
           =TT | n+k-j∑y ∈ C_j ∑ξ(x) ν(y) 
             +  T ∤ n+k-j∑y ∈ C_j ∑ξ(x) ν(y) e^iπ(n+k-j)(T-1)/Tsin(π(n+k-j))/sin(π(n+k-j)/T)_=0
       =TT | n+k-j∑y ∈ C_j∑ξ(x) ν(y) > 0
    




 §.§ Quasi-ergodic distribution for the classical non-moving sense in the general case


Now assume that the sub-transition matrix Q is not necessarily irreducible. For each x ∈{1, …, K}, we denote by D_x the subset of {1, …, K} defined by

    D_x := {y ∈{1, …, K} : ∃ n,m ∈, _x(Y_n = y) > 0  and _y(Y_m = x) > 0}

It is well-known that (D_x)_x ∈{1, …, K} are equivalence classes. Note that,
for each x, the restriction of Y on D_x is irreducible. Thus we can associate, for each D_x, a period T_x.  We can also associate to D_x a spectral radius ρ_x and some left and right-eigenvectors (v_x,l)_0 ≤ l ≤ T_x-1 and (w_x,l)_0 ≤ l ≤ T_x-1 constructed in the same way as in the subsection <ref>. Particularly, for every x ∈{1, …, K}, ν_x := v_x,0 and ξ_x := w_x,0 are vectors whose all the components are positive and such that <ν_x,>=<ν_x,ξ_x>=1. We define also, for any x, 
    φ_x : f →j=1|D_x|∑ f(j)ν_x(j) ξ_x(j)
 
where |D_x| is the number of elements in D_x. 
Now fix μ∈ M_1({1, …, K}). Denote by Supp(μ) the support of μ. Then we can define 
    
    B = {x ∈{1, …, K }: Supp(μ) ∩ D_x ∅}

    
    ρ_max = x ∈ Bmax ρ_x

 and we define B_max as follows

    B_max = {x ∈ B : ρ_x = ρ_max}

We set the following hypothesis


There exists x_max∈{1, …, K} such that

    B_max = D_x_max

Under this hypothesis, the following notation will be used

    ν_max = ν_x_max


    ξ_max = ξ_x_max


    φ_max = φ_x_max


In all what follows, we have to keep in mind that the definition of B_max implicitly depends on the initial distribution μ (more precisely on the support of μ). 

The following statement explains therefore that the quasi-ergodic distribution exists if the Hypothesis <ref> holds.


Let μ∈ M_1({1, …, K}). Then, if the Hypothesis <ref> holds, the following convergence holds for any measurable bounded function f :{1, …, K}→,

    _μ(1/nk=0n-1∑ f(Y_k) | τ > n) n →∞⟶φ_max(f)




According to Proposition <ref>, giving the fact that Y is irreducible on each D_x, we have for any x ∈{1, …, K}

    _x(1/nk=0n-1∑ f(Y_k) _τ > n) = ρ_x^n φ_x(f) l=0T-1∑ e^-2inlπ/T_x  <w_x,l,δ_x> <v_x,l,> + o(ρ_x^n)

Thus, for any μ∈ M_1(E)

    _μ(1/nk=0n-1∑ f(Y_k) | τ > n) = j=1K∑μ(j) _j(1/nk=0n-1∑ f(Y_k) _τ > n)/j=1K∑μ(j) _j(τ > n)
       =j=1K∑μ(j) ρ_j^n φ_j(f) l=0T_j-1∑ e^-2inlπ/T_j  <w_j,l,δ_x> <v_j,l,> + o(ρ_j^n)/j=1K∑μ(j) ρ_j^n l=0T_j-1∑ e^-2inlπ/T_j  <w_j,l,δ_x> <v_j,l,> + o(ρ_j^n)
       =j ∈ B_max∑φ_j(f)μ(j) l=0T_j-1∑ e^-2inlπ/T_j  <w_j,l,δ_x> <v_j,l,> + o(1)/j ∈ B_max∑μ(j) l=0T_j-1∑ e^-2inlπ/T_j  <w_j,l,δ_x> <v_j,l,> + o(1)
       =φ_max(f)j ∈ B_max∑μ(j) l=0T_j-1∑ e^-2inlπ/T_j  <w_j,l,δ_x> <v_j,l,> + o(1)/j ∈ B_max∑μ(j) l=0T_j-1∑ e^-2inlπ/T_j  <w_j,l,δ_x> <v_j,l,> + o(1)

where the Hypothesis <ref> was used for the last equality, implying that φ_j(f) = φ_max(f) for all j ∈_max. Note moreover that this hypothesis is useful only to make this equality right. 

Then using Proposition <ref>, we can conclude

    _μ(1/nk=0n-1∑ f(Y_k) | τ > n) n →∞⟶φ_max(f)





 §.§ Quasi-ergodic distribution with periodic moving killing subset

In this subsection we are interested in the quasi-ergodicity of the chain X defined in the Introduction considering that the boundaries are moving γ-periodically.
We denote by Y = (Y_n)_n ∈ the Markov chain defined on E ×/γ by 

    Y_n = (X_n,n)

where n̅ is the residue of n, modulo γ. 
Y is therefore a Markov chain defined on a finite space state, which is irreducible  if and only if gcd(T(X),γ) = 1, where T(X) is the period of (X_n)_n ∈. If the chain Y is actually irreducible, the associated period is  
    T = LCM(T(X),γ)

where LCM(·, ·) refers to the least common multiple.
 Moreover we have

    τ = inf{n ≥ 0 : X_n ∈ A_n} = inf{n ≥ 0 : Y_n ∈∂}

with 
    ∂ := {(x,k): x ∈ A_k}

 Remark that ∂ is a non moving killing subset for the chain Y. Thus we can apply Theorem <ref> to the process Y which yields the following theorem


Let μ∈ M_1(E_0). Assume that (A_n)_n ∈ is periodic and  Y defined in (<ref>) satisfies the Hypothesis <ref>. Then, for any measurable bounded function f,

    _μ(1/nk=0n-1∑ f(X_k) | τ > n) n →∞⟶(x,y) ∈ E ×/γ -∑ f(x) ν_max(x,y) ξ_max(x,y)

where ν_max and ξ_max are the probability measures defined in (<ref>) and (<ref>) relatively to Y.

We can also give the following corollary which requires assumptions on X and (A_n)_n ∈.


Assume that (A_n)_n ∈ is γ-periodic and that gcd(T,γ)=1 (where T is the period of X). Then there exists η∈ M_1(E) such that, for any μ∈ M_1(E_0) and any f bounded measurable,
   
    _μ(1/nk=0n-1∑ f(X_k) | τ > n) n →∞⟶∫ f dη



It is enough to apply Theorem <ref> to the chain Y defined on (<ref>) and to deduce the results on X thanks to the following equality

    𝔼_μ(1/nk=0n-1∑ f(X_k) | τ > n) = 𝔼_μ⊗δ_0(1/nk=0n-1∑f̃(Y_k) | τ > n),   ∀ n ≥ 1

where f̃ is the projection on the first component.




§ EXISTENCE OF Q-PROCESS WITH BOUNDARIES MOVING PERIODICALLY

In this section, we are interested in the Q-process, which can be interpreted as the law of the process X conditioned never to be killed by the moving boundary. As before, we still consider that the boundary moves periodically period γ. 

To show the existence of the Q-process, we will consider again the Markov chain Y defined in (<ref>), that is defined by

    Y_n = (X_n,n),  ∀ n ∈

and we take back the notation introduced in subsection <ref> associated to Y. 

The following statement ensures the existence of a Q-process even when the boundary moves. However, it is interesting to observe that we lose the homogeneity of the Q-process because of the movement of the killing boundary.


For any n ∈ and any x ∈ E_0, the probability measure ℚ_x defined by

    ℚ_x(X_1 ∈·, …, X_n ∈·) = m →∞lim_x(X_1 ∈·, …, X_n ∈· | τ > m)

is well-defined and, under the probability ℚ_x, (X_n)_n ∈ is a time-inhomogeneous Markov chain such that for any n ∈, for any (y,z) ∈ E_n-1× E_n

    ℚ_x(X_n = z | X_n-1 = y) = ξ_x(z,n)/ρ_x ξ_x(y,n-1)_y(X_1 = z, τ_A_n > 1)









For any m,n ∈, for any f_1, …, f_n measurable bounded functions and for any x ∈ E_0,

    _x(f_1(Y_1) … f_n(Y_n) | τ > n+m)    = _x(f_1(Y_1) … f_n(Y_n) _τ > n+m)/_x(τ > n+m)
       =_x(f_1(Y_1) … f_n(Y_n) _τ > n_Y_n(τ > m)/_x(τ > n+m))

According to the equality (<ref>) applied to the function equal to 1, for any y ∈ E × Z/γ - $̣ andn ∈,_y(τ > n) = ρ_y^n l=0T_y-1∑ e^-2inlπ/T_y <w_y,l, δ_y> <v_y,l,> + o(ρ_y^n)Thus, using this in (<ref>),

    _x(f_1(Y_1) … f_n(Y_n)| τ > m+n) 
       = _x(f_1(Y_1)… f_n(Y_n) _τ > nρ_Y_n^m l=0T_Y_n-1∑ e^-2imlπ/T_Y_n <w_Y_n,l,δ_Y_n> <v_Y_n,l,> + o(ρ_Y_n^m)/ρ_x^n+ml=0T_x-1∑ e^-2i(n+m)lπ/T_x <w_x,l,δ_x> <v_x,l,> + o(ρ_x^n+m)) 
       = _x(f_1(Y_1)… f_n(Y_n) _τ > nρ_x^m l=0T_x-1∑ e^-2imlπ/T_x <w_x,l,δ_Y_n> <v_x,l,> + o(ρ_x^m)/ρ_x^n+ml=0T_x-1∑ e^-2i(n+m)lπ/T_x <w_x,l,δ_x> <v_x,l,> + o(ρ_x^n+m)) 
       = _x(f_1(Y_1)… f_n(Y_n) _τ > nl=0T_x-1∑ e^-2imlπ/T_x <w_x,l,δ_Y_n> <v_x,l,> + o(1)/ρ_x^nl=0T_x-1∑ e^-2i(n+m)lπ/T_x <w_x,l,δ_x> <v_x,l,> + o(ρ_x^n))

The passage from (<ref>) to (<ref>) is due to the fact that, for anyn ∈,Y_n ∈D_xalmost surely and the quantitiesT_x,ρ_x,w_x,landv_x,ldepends only onD_x. 

Since the restriction of the chainYonD_xis irreducible, we can construct as in the subsection <ref> some clusters(C_j)_0 ≤j ≤T_x-1such thatx ∈C_0and

    _x(Y_k+nT_x∈ C_k, τ > k+nT_x) = 1,   ∀ k ∈{0, …, T_x-1}, ∀ n ∈
 
For anyy ∈D_x, denote byj(y)the integer such thaty ∈C_j(y). Then we deduce from the equality (<ref>) in the subsection <ref> that for anyy ∈E ×/γ- $̣ and n ∈, 

    e^-2inlπ/T_x <w_x,l,δ_y> = e^-2iπ(n+j(y))l/T_xξ_x(y)

Thus, according to (<ref>) and the previous equality,

    _x(f_1(Y_1) … f_n(Y_n)| τ > m+n) 
       = _x(f_1(Y_1)… f_n(Y_n) _τ > nξ_x(Y_n) (l=0T_x-1∑ e^-2iπ(m+j(Y_n))l/T_x<v_x,l,> + o(1))/ρ_x^nξ_x(x) (l=0T_x-1∑ e^-2iπ(m+n+j(x))l/T_x<v_x,l,> + o(1)))

However, for any n ∈, 

    j(Y_n) = j(x) + n  T_x,   a.s.

and for any m,n ∈,

    l=0T_x-1∑ e^-2iπ(m+n+j(x))l/T_x<v_x,l,>  0
Since the state space E × / γ is finite, we may first consider function f_i(y) = _y = x_i, so that quantities in the ratio except _τ > n are fixed. This justifies that we can exchange the expectation and the limit as n →∞ in the previous expression.
We deduce that,

    _x(f_1(Y_1) … f_n(Y_n)| τ > m+n) m →∞⟶_x(f_1(Y_1)… f_n(Y_n) _τ > nξ_x(Y_n)/ρ_x^nξ_x(x,0))

The statement on X is obtained using projection functions and we can deduce from it the transition kernel of the Q-process.




§ EXAMPLE : DISCRETE-TIME RANDOM WALK


We shall illustrate the  previous results by looking at a discrete-time random walk.
Let p ∈ ]0,1[. We denote by (M_n^p)_n ∈ℕ the Markov chain defined on  such that

    (M_n+1^p = M_n^p + 1 | M_n^p) = 1-p 
       (M_n+1^p = M_n^p - 1 | M_n^p) = p

Before dealing with the quasi-ergodicity with moving boundaries, let us recall some properties about quasi-stationarity concerning random walks. For any K ≥ 1 we define

    T_K = inf{n ≥ 0 : M_n^p ∈ (-∞,0] ∪ [K+1, ∞) }

The sub-Markovian transition matrix associate to (M_n  T_K^p)_n ∈ is the matrix Q_K∈ M_K(ℝ) defined by :  

    Q_K = [   0 1-p   0   …   0   0;   p   0 1-p   …   0   0;   0   p   0   …   0   0;   ⋮   ⋮   ⋮   ⋱   ⋮   ⋮;   0   0   0   …   0 1-p;   0   0   0   …   p   0 ]

For any K ≥ 1, denote by P_K(X) the characteristic polynomial of Q_K. Using standard algebraic manipulations, one can show that for any K ≥ 1, the following recurrence relation is satisfied 

    P_K+2(X) = -X P_K+1(X) - p(1-p) P_K(X)

with P_1(X) = -X and P_2(X) = X^2 - p(1-p). We set P_0(X) = 1.

For any K ≥ 0, define

    U_K(X) = (-1/√(p(1-p)))^K P_K(2√(p(1-p))X)

Then the following equation is satisfied

    U_K+2(X) = 2X U_K+1(X) - U_K(X)

for which U_0(X) = 1 and U_1(X) = 2X. In other words, the sequence (U_K)_K ≥ 0 are the Chebyshev's polynomials of the second kind and we have for any θ∈ℝ

    U_K(cos(θ)) = sin((K+1)θ)/sin(θ)

The set of roots of U_K, hence of P_K, is thus well-known. It follows  

    Sp(Q_K) = {λ_j := 2√(p(1-p))cos(j π/K+1) : j ∈{1, …, K}}

We are interested now in the eigenvectors of Q_K. 

Let K ≥ 1. Then, for any j ∈{1, …, K}, Ker(Q_K - λ_j I_k) = Span(x_j) where

    x_j(i) = (-1/1-p)^i-1 P_i-1(λ_j) = (√(p/1-p))^i-1sin(ijπ/K+1)/sin(jπ/K+1),   ∀ i ∈{1, … , K}



Let λ∈ Sp(Q_K). We want to find all the eigenvectors x = (x(i))_1 ≤ i ≤ K associated to λ such that x(1)=1. We will prove the proposition by double induction. 


Base case: According to the relation Q_K x = λ x, we have 

    λ x(1) = (1-p) x(2)

Having x(1)=1, we will have therefore x(2) = λ/1-p = -1/1-p P_1(λ), which conclude the base case


Inductive step: Let i ∈{3, …, K-1}. We assume that the equality is satisfied for i-1 and i-2, so we have 

    x(i-2) = (-1/1-p)^i-3 P_i-3(λ) 
       x(i-1) = (-1/1-p)^i-2 P_i-2(λ)

Using λ x = Q_K x, 

    λ x(i-1) = px(i-2) + (1-p)x(i)

So

    x(i)    = 1/1-p( λ x(i-1) - px(i-2)) 
       =1/1-p( λ(-1/1-p)^i-2 P_i-2(λ) - px(i-2)) 
       = (-1/1-p)^i-1(-λ P_i-2(λ) - p(1-p) P_i-3(λ)) 
       =(-1/1-p)^i-1 P_i-1(λ)

which concludes the proof.

The previous proposition gives us left and right eigenvectors of Q_K : if we denote by (v_i)_1 ≤ i ≤ K (respectively (w_i)_1 ≤ i ≤ K) the left (respectively right) eigenvectors satisfying v_iQ_K = λ_i v_i (respectively Q_K w_i = λ_i w_i), then

    v_i(j) = (√(1-p/p))^j-1sin(ijπ/K+1)/sin(iπ/K+1)
       w_i(j) = (√(p/1-p))^j-1sin(ijπ/K+1)/sin(iπ/K+1)

In particular, considering the spectral radius λ_1, the quasi-stationary distribution ν and the right-eigenvector ξ associated to λ_1 satisfying <ν,ξ>=1 are as follows:

    ν(j) = (√(1-p/p))^j-1sin(jπ/K+1)/k=1K∑(√(1-p/p))^k-1sin(kπ/K+1)
       ξ(j) = k=1K∑(√(1-p/p))^k-1sin(kπ/K+1)/k=1K∑sin^2(kπ/K+1)(√(p/1-p))^j-1sin(jπ/K+1)

We are interested now in moving boundaries. Let N ≥ 1  and consider the simplest case where (A_n)_n ∈ is defined by

    A_n = {[   (-∞,0] ∪ [2N,∞)      if n is even; (-∞,1] ∪ [2N-1,∞)       if n is odd ].

Recall the previous notation 

    Y^p_n = (M^p_n τ_0,n)
   
with n∈/2. The chain is not irreducible (if M^p_0 is even, then for any n, M^p_n have the same parity as n). It admits exactly two irreducible subsets:

    
  * P = {(x,y) ∈ E : x+y  is even}
    
  * I = {(x,y) ∈ E : x+y  is odd}

But, as we can see in Figure <ref>, the chain Y^p behaves as a random walk on each irreducible subsets:


  * On P, Y^p has the same behavior as a random walk on ℤ starting from [2,2N-2] absorbed by {1,2N-1}.

  * On I, Y^p has the same behavior as a random walk on ℤ starting from [1,2N-1] absorbed by {0,2N}.

Denote by Y^p_ P (respectively Y^p_ I) the Markov chain such that for any μ∈ M_1( P) (respectively M_1( I))

    _μ(Y^p_1 ∈·) = _μ((Y^p_ P)_1 ∈·)  (respectively _μ(Y^p_1 ∈·) = _μ((Y^p_ I)_1 ∈·) )

Let μ∈ M_1(E ×/2). Then there are λ∈ [0,1] and μ_ P, μ_ I∈ M_1( P) × M_1( I) such that

    μ = λμ_ P + (1-λ)μ_ I

Hence we see that two cases are possible



  * if λ = 1, B_max =  P. Then ρ_max = 2√(p(1-p))cos(π/2(N-1)),  and

    _μ(1/nk=0n-1∑ f(M^p_k) | τ > n) n →∞⟶j=22N-2∑ f(j) sin^2((j-1)π/2(N-1))/k=12N-3∑sin^2(kπ/2(N-1))
 


  * if λ 1, B_max =  I. Then ρ_max = 2√(p(1-p))cos(π/2N),  and

    _μ(1/nk=0n-1∑ f(M^p_k) | τ > n) n →∞⟶j=12N-1∑ f(j) sin^2(jπ/2N)/k=12N-1∑sin^2(kπ/2N)



