The Number of Generations Entirely Visited for Recurrent Random Walks in a Random Environment

Abstract

In this paper we deal with a random walk in a random environment on a super-critical Galton–Watson tree. We focus on the recurrent cases already studied by Hu and Shi (Ann. Probab. 35:1978–1997, 2007; Probab. Theory Relat. Fields 138:521–549, 2007), Faraud et al. (Probab. Theory Relat. Fields, 2011, in press), and Faraud (Electron. J. Probab. 16(6):174–215, 2011). We prove that the largest generation entirely visited by these walks behaves like logn, and that the constant of normalization, which differs from one case to another, is a function of the inverse of the constant of Biggins’ law of large numbers for branching random walks (Biggins in Adv. Appl. Probab. 8:446–459, 1976).

Appendix A

In this appendix, for completeness, we describe and sketch the proof of some classical results. Given a vertex $x\in\mathbb{T}$, we denote by x ₀:=ϕ,…,x _n :=x the vertices on 〚ϕ,x〛 with |x _i |=i for all 0≤i≤n.

1.1 A.1 Biggins–Kyprianou Identities

For any n≥1 and any measurable function F:ℝⁿ×ℝⁿ→[0,+∞), the Biggins–Kyprianou identity is given by

$$ E \biggl[\sum_{\vert x\vert=n}e^{-V(x)-\psi(1)n}F \bigl(V(x_i),1\leq i\leq n\bigr) \biggr]=E\bigl[F(S_i,1 \leq i\leq n)\bigr] $$

(A.1)

where (S _i −S _i−1) _i≥1, are i.i.d. random variables, and the distribution of S ₁ is determined by

$$ E\bigl[f(S_1)\bigr]=E \biggl[\sum _{\vert x\vert=1}e^{-V(x)-\psi (1)}f\bigl(V(x)\bigr) \biggr], $$

(A.2)

for any measurable function f:ℝ→[0,+∞). A proof can be found in [2]; see also [14].

1.2 A.2 Classical Results About Birth and Death Chains

Lemma A.6

For x′∈〚ϕ,x〛:

(A.3)

(A.4)

where the $x^{\prime}_{x}$ are the only children of x′ in 〚x′,x〛.

Proof

Let (σ _n ) _n≥0 be the family of stopping times defined by and define $Z_{n}=X_{\sigma_{n}}$ for n≥0. (Z _n ) _n≥0 is a birth and death Markov chain on 〚ϕ,x〛 with transition probabilities given by

∀1≤i≤n−1 and p _ϕ =q _x =1. Indeed

Let us introduce

$$\xi_{0}:=1,\qquad\xi_{\ell}:=\prod_{k=1}^{\ell}\frac{q_{k}}{p_{k}},\quad \ell\geq1, $$

and consider f:ℕ→ℝ given by f(ϕ)=0; for $1\leq k\leq n,f(x_{k})=\sum_{\ell=0}^{k-1}\xi_{\ell}$. We easily see that (f(Z _k )) _k≥0 is a martingale. With τ _i =inf{m≥,0,Z _m =x _i } and for 1≤i<j<k, according to the Optional Stopping Time Theorem, for 1≤i<j<k:

recalling that . Since {τ _x <τ _x′}={T _x <T _x′} conditionally on {X ₀=x′ _x }, formula (A.3) is proved. □

1.3 A.3 About (γ _n ,n)

Let us define

$$ \gamma_n(x):=\left\{ \begin{array}{@{}l@{\quad}l} 0&\mbox{if $\vert x\vert=n$,}\\[2mm] \frac{{1}/{p(x,\overset{\leftarrow}{x})}+\sum _{i=1}^{N_x}A(x^i)\gamma_n(x^i)}{1+\sum_{i=1}^{N_x}A(x^i)\beta _n(x^i)},&\mbox{if $1\leq\vert x\vert<n$,}\\[3mm] \sum_{i=1}^Np(\phi,\phi_i)\gamma_n(\phi_i), &\mbox{if $x=\phi$}. \end{array} \right. $$

(A.5)

Lemma A.7

Assuming ψ(1)=0:

$$ \sup_{n \geq1} \frac{\gamma_n(\phi)}{n}< + \infty,\quad\mathbb{P}\ \mathrm{a.s.} $$

(A.6)

This result is already proved in the case of a b-ary tree (see for instance [6]). Here, we treat the case of a Galton–Watson tree.

Proof

First, observe that for all 2≤k≤n:

(A.7)

where K is a constant satisfying $\forall x\in\mathbb {T},p(x,\overset {\leftarrow}{x})^{-1}\leq K$. The existence of K is provided by assumptions (1.1).

As p(ϕ,ϕ ⁱ)≤A(ϕ ⁱ), ∀1≤i≤N, we deduce from (A.5):

$$ \gamma_n(\phi)\leq\sum_{i=1}^NA \bigl(\phi^{i}\bigr)\gamma_n\bigl(\phi^{i}\bigr), $$

(A.8)

and note that formula (A.5) implies

$$ \gamma_n(x)\leq K+\sum_{i=1}^{N_x}A \bigl(x^{i}\bigr)\gamma_n\bigl(x^{i}\bigr),\quad \forall 1\leq\vert x\vert\leq n. $$

(A.9)

Then from (A.8) and (A.9), we deduce formula (A.7) for k=2:

Assume that (A.7) is true for a certain k≥2 ; we prove that it still true for k+1. Using again (A.9):

Applying formula (A.7) to k=n and recalling that γ _n (x)=0 for |x|=n:

(A.10)

where M _j :=∑_|x|=j ∏_〛ϕ,x〛 A(y). (M _j ) _j≥1 is a positive $\mathcal{F}_{j}$-martingale with M ₀=1 and $\mathcal{F}_{j}:=\sigma\{(A(x^{1}),\ldots, A(x^{N_{x}}),N_{x}): \vert x\vert\leq j,x\in\mathbb{T}\}$. We now may state:

• obviously we have positivity and for all j≥0, $M_{j}\in \mathcal{F}_{j}$;

• for all x∈T, as $(A(x^{1}),\ldots, A(x^{N_{x}}),N_{x})$ is equal in law to the vector (A ₁,…,A _N ,N):

  $$\mathbb{E}[M_{j+1}\vert\mathcal{F}_j] =M_j \mathbb{E}\Biggl[\sum_{i=1}^{N}A_i \Biggr], $$

  and we conclude with $M_{0}=\mathbb{E}[\sum_{i=1}^{N}A_{i}]=1$, since ψ(1)=0.

Consequently, there exists an almost sure limit for (M _j ) _j≥0, which implies that sup _j M _j <∞ almost surely.

Thus, (A.10) implying $\frac{\gamma_{n}(\phi)}{n}\leq K \sup _{j}M_{j}$, we see that the proof is complete. □