Zhiqiang Gao

Quansheng Liu
email: quansheng.liu@univ-ubs.fr

Exact convergence rates in central limit theorems for a branching random walk with a random environment in time *

Keywords: 2000 Mathematics Subject Classification. Preliminary 60K37, 60J10, 60F05, 60J80 Branching random walk, random environment in time, central limit theorems, convergence rate

derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process. We extend Chen's results to a branching random walk under weaker moment conditions. For the branching Wiener process, our results sharpen Chen's by relaxing the second moment condition used by Chen to a moment condition of the form EX(ln + X) 1+λ < ∞. In the rate functions that we find for a branching random walk, we figure out some new terms which didn't appear in Chen's work. The results are established in the more general framework, i.e. for a branching random walk with a random environment in time. The lack of the second moment condition for the offspring distribution and the fact that the exponential moment does not exist necessarily for the displacements make the proof delicate; the difficulty is overcome by a careful analysis of martingale convergence using a truncating argument. The analysis is significantly more awkward due to the appearance of the random environment.

Introduction

The theory of branching random walk has been studied by many authors. It plays an important role, and is closely related to many problems arising in a variety of applied probability setting, including branching processes, multiplicative cascades, infinite particle systems, Quicksort algorithms and random fractals (see e.g. [START_REF] Liu | Fixed points of a generalized smoothing transformation and applications to branching processes[END_REF][START_REF] Liu | On generalized multiplicative cascades[END_REF]). For recent developments of the subject, see e.g. Hu and Shi [START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF], Shi [START_REF] Shi | Branching random walks. Saint-Flour's summer course[END_REF], Hu [START_REF] Hu | How big is the minimum of a branching random walk?[END_REF], Attia and Barral [START_REF] Attia | Hausdorff and packing spectra, large deviations and free energy for branching random walks in R d[END_REF] and the references therein.

In the classical branching random walk, the point processes indexed by the particles u, formulated by the number of its children and their displacements, have a fixed constant distribution for all particles u. In reality this distributions may vary from generation to generation according to a random environment, just as in the case of a branching process in random environment introduced in [START_REF] Athreya | On branching processes with random environments. I. Extinction probabilities[END_REF][START_REF] Athreya | Branching processes with random environments. II. Limit theorems[END_REF][START_REF] Smith | On branching processes in random environments[END_REF]. In other words, the distributions themselves may be realizations of a stochastic process, rather than being fixed. This property makes the model be closer to the reality compared to the classical branching random walk. In this paper, we shall consider such a model, called a branching random walk with a random environment in time .

Different kinds of branching random walks in random environments have been introduced and studied in the literature. Baillon, Clément, Greven and den Hollander [START_REF] Baillon | A variational approach to branching random walk in random environment[END_REF][START_REF] Greven | Branching random walk in random environment: phase transitions for local and global growth rates[END_REF] considered the case where the offspring distribution of a particle situated at z ∈ Z d depends on a random environment indexed by the location z, while the moving mechanism is controlled by a fixed deterministic law. Comets and Popov [START_REF] Comets | On multidimensional branching random walks in random environment[END_REF][START_REF] Comets | Shape and local growth for multidimensional branching random walks in random environment[END_REF] studied the case where both the offspring distributions and the moving laws depend on a random environment indexed by the location. In the model studied in [START_REF] Birkner | Branching processes in random environment-a view on critical and subcritical cases[END_REF][START_REF] Comets | Branching random walks in space-time random environment: survival probability, global and local growth rates[END_REF][START_REF] Hu | Localization for branching random walks in random environment[END_REF][START_REF] Nakashima | Almost sure central limit theorem for branching random walks in random environment[END_REF][START_REF] Yoshida | Central limit theorem for branching random walks in random environment[END_REF], the offspring distribution of a particle of generation n situated at z ∈ Z d (d ≥ 1) depends on a random space-time environment indexed by {(z, n)}, while each particle performs a simple symmetric random walk on d-dimensional integer lattice Z d (d ≥ 1).

The model that we study in this paper is different from those mentioned above. It should also be mentioned that recently another different kind of branching random walks in time inhomogeneous environments has been considered extensively, see e.g. Fang and Zeitouni (2012, [16]), Zeitouni (2012, [START_REF] Zeitouni | Branching random walks and Gaussian fields[END_REF]) and Bovier and Hartung(2014, [10]). The readers may refer to these articles and references therein for more information.

Denote by Z n (•) the counting measure which counts the number of particles of generation n situated in a given set. For the classical branching random walk, a central limit theorem on Z n (•), first conjectured by Harris (1963, [20]), was shown by Asmussen and Kaplan (1976, [1, 25]), and then extended to a general case by Klebaner (1982, [26]) and Biggins (1990, [7]); for a branching Wiener process, Révész (1994,[34]) studied the convergence rates in the central limit theorems and conjectured the exact convergence rates, which were confirmed by Chen (2001,[11]). Kabluchko (2012,[40]) generalized Chen's partial results using a different method. Révész, Rosen and Shi (2005, [START_REF] Révész | Large-time asymptotics for the density of a branching Wiener process[END_REF]) obtained a large time asymptotic expansion in the local limit theorem for branching Wiener processes, generalizing Chen's result.

The first objective of our present paper is to extend Chen's results to the branching random walk under weaker moment conditions. In our results about the exact convergence rate in the central limit theorem and the local limit theorem, the rate functions that we find include some new terms which didn't appear in Chen's paper [START_REF] Chen | Exact convergence rates for the distribution of particles in branching random walks[END_REF]. In Chen's work, the second moment condition was assumed for the offspring distribution. Although the setting we consider now is much more general, in our results the second moment condition will be relaxed to a moment condition of the form EX(ln + X) 1+λ < ∞ . It has been well known that in branching random walks, such a relaxation is quite delicate. Another interesting aspect is that we do not assume the existence of exponential moments for the moving law, which holds automatically in the case of the branching Wiener process. The lack of the second moment condition (resp. the exponential moment condition) for the offspring distribution (resp. the moving law) makes the proof delicate. The difficulty will be overcome via a careful analysis of the convergence of some associated martingales using truncating arguments.

The second objective of our paper is to extend the results to the branching random walk with a random environment in time. This model first appeared in Biggins and Kyprianou (2004, [8, Section 6]), where a criterion was given for the non-degeneration of the limit of the natural martingale; see also Kuhlbusch (2004, [27]) for the equivalent form of the criterion on weighted branching processes in random environment. For Z n (•) and related quantities on this model, Liu (2007,[31]) surveyed several limit theorems, including large deviations theorems and a law of large numbers on the rightmost particle. In [START_REF] Gao | Central limit theorems for a branching random walk with a random environment in time[END_REF], Gao, Liu and Wang showed a central limit theorem on the counting measure Z n (•) with appropriate norming. Here we study the convergence rate in the central limit theorem and a local limit theorem for Z n (•). Compared with the classical branching random walk, the approach is significantly more difficult due to the appearance of the random environment.

The article is organized as follows. In Section 2, we give a rigorous description of the model and introduce the basic assumptions and notation, then we formulate our main results as Theorems 2.3 and 2.4. In Section 3, we introduce some notation and recall a theorem on the Edgeworth expansions for sums of independent random variables used in our proofs. We give the proofs of the main theorems in Section 5 and 6, respectively. Whilst Section 4 will be devoted to the proofs of the reminders.

2 Description of the model and the main results

Description of the model

We describe the model as follows ( [START_REF] Gao | Central limit theorems for a branching random walk with a random environment in time[END_REF][START_REF] Liu | Branching random walks in random environment[END_REF]). A random environment in time ξ = (ξ n ) is formulated as a sequence of random variables independent and identically distributed with values in some measurable space (Θ, F ). Each realization of ξ n corresponds to two probability distributions: the offspring distribution

p(ξ n ) = (p 0 (ξ n ), p 1 (ξ n ), • • • ) on N = {0, 1, • • • },
and the moving distribution G(ξ n ) on R. Without loss of generality, we can take ξ n as coordinate functions defined on the product space (Θ N , F ⊗N ) equipped with the product law τ of some probability law τ 0 on (Θ, F ), which is invariant and ergodic under the usual shift transformation θ on

Θ N : θ(ξ 0 , ξ 1 , • • • ) = (ξ 1 , ξ 2 , • • • ).
When the environment ξ = (ξ n ) is given, the process can be described as follows. It begins at time 0 with one initial particle ∅ of generation 0 located at S ∅ = 0 ∈ R; at time 1, it is replaced by

N = N ∅ new particles ∅i = i(1 ≤ i ≤ N ) of generation 1, located at S i = L ∅i (1 ≤ i ≤ N ),
where N, L 1 , L 2 , • • • are mutually independent, N has the law p(ξ 0 ), and each L i has the law G(ξ 0 ). In general, each particle u = u 1 ...u n of generation n is replaced at time n + 1 by N u new particles ui(1 ≤ i ≤ N u ) of generation n + 1, with displacements L ui (1 ≤ i ≤ N u ), so that the i-th child ui is located at

S ui = S u + L ui ,
where N u , L u1 , L u2 , • • • are mutually independent, N u has the law p(ξ n ), and each L ui has the same law G(ξ n ). By definition, given the environment ξ, the random variables N u and L u , indexed by all the finite sequences u of positive integers, are independent of each other.

For each realization ξ ∈ Θ N of the environment sequence, let (Γ, G, P ξ ) be the probability space under which the process is defined (when the environment ξ is fixed to the given realization). The probability P ξ is usually called quenched law. The total probability space can be formulated as the product space (Θ N × Γ, E N ⊗ G, P), where P = E(δ ξ ⊗ P ξ ) with δ ξ the Dirac measure at ξ and E the expectation with respect to the random variable ξ, so that for all measurable and positive g defined on Θ N × Γ, we have

Θ N ×Γ g(x, y)dP(x, y) = E Γ g(ξ, y)dP ξ (y).
The total probability P is usually called annealed law. The quenched law P ξ may be considered to be the conditional probability of P given ξ. The expectation with respect to P will still be denoted by E; there will be no confusion for reason of consistence. The expectation with respect to P ξ will be denoted by E ξ .

Let T be the genealogical tree with {N u } as defining elements. By definition, we have:

(a) ∅ ∈ T; (b) ui ∈ T implies u ∈ T; (c) if u ∈ T, then ui ∈ T if and only if 1 ≤ i ≤ N u . Let T n = {u ∈ T : |u| = n}
be the set of particles of generation n, where |u| denotes the length of the sequence u and represents the number of generation to which u belongs.

Main results

Let Z n (•) be the counting measure of particles of generation n: for B ⊂ R,

Z n (B) = u∈Tn 1 B (S u ).
Then {Z n (R)} constitutes a branching process in a random environment (see e.g. [START_REF] Athreya | On branching processes with random environments. I. Extinction probabilities[END_REF][START_REF] Athreya | Branching processes with random environments. II. Limit theorems[END_REF][START_REF] Smith | On branching processes in random environments[END_REF]). For n ≥ 0, let N n (resp. L n ) be a random variable with distribution p(ξ n ) (resp. G(ξ n )) under the law P ξ , and define

m n = m(ξ n ) = E ξ N n , Π n = m 0 • • • m n-1 , Π 0 = 1.
It is well known that the normalized sequence

W n = 1 Π n Z n (R), n ≥ 1
constitutes a martingale with respect to the filtration (F n ) defined by

F 0 = {∅, Ω}, F n = σ(ξ, N u : |u| < n), for n ≥ 1.
Throughout the paper, we shall always assume the following conditions:

E ln m 0 > 0 and E 1 m 0 N 0 ln + N 0 1+λ < ∞, (2.1)
where the value of λ > 0 is to be specified in the hypothesis of the theorems. Under these conditions, the underlying branching process {Z n (R)} is supercritical, Z n (R) → ∞ with positive probability, and the limit W = lim n W n verifies EW = 1 and W > 0 almost surely (a.s.) on the explosion event {Z ∞ → ∞} (cf. e.g. [START_REF] Athreya | Branching processes with random environments. II. Limit theorems[END_REF][START_REF] Tanny | A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means[END_REF]).

For n ≥ 0, define

l n = E ξ L n , σ (ν) n = E ξ L n -l n ν , for ν ≥ 2; ℓ n = n-1 k=0 l k , s (ν) n = n-1 k=0 σ (ν) k , for ν ≥ 2, s n = s (2) n 1 2 .
We will need the following conditions on the motion of particles:

P lim sup |t|→∞ E ξ e it L0 < 1 > 0 and E | L 0 | η < ∞, (2.2)
where the value of η > 1 is to be specified in the hypothesis of the theorems. The first hypothesis means that Cramér's condition about the characteristic function of L 0 holds with positive probability.

Let {N 1,n } and {N 2,n } be two sequences of random variables, defined respectively by

N 1,n = 1 Π n u∈Tn (S u -ℓ n ) and N 2,n = 1 Π n u∈Tn (S u -ℓ n ) 2 -s 2 n .
We shall prove that they are martingales with respect to the filtration (D n ) defined by

D 0 = {∅, Ω}, D n = σ(ξ, N u , L ui : i ≥ 1, |u| < n), for n ≥ 1.
More precisely,we have the following propositions.

Proposition 2.1. Assume (2.1) and E ln -m 0 1+λ < ∞ for some λ > 1, and

E | L 0 | η < ∞ for some η > 2.
Then the sequence {(N 1,n , D n )} is a martingale and converges a.s.:

V 1 := lim n→∞ N 1,n exists a.s. in R.
Proposition 2.2. Assume (2.1) and E ln -m 0 1+λ < ∞ for some λ > 2 , and

E | L 0 | η < ∞ for some η > 4.
Then the sequence {(N 2,n , D n )} is a martingale and converges a.s.:

V 2 := lim n→∞ N 2,n exists a.s. in R.
Our main results are the following two theorems. The first theorem concerns the exact convergence rate in the central limit theorem about the counting measure Z n , while the second one is a local limit theorem. We shall use the notation

Z n (t) = Z n ((-∞, t]), φ(t) = 1 √ 2π e -t 2 /2 , Φ(t) = t -∞ φ(x)dx, t ∈ R. Theorem 2.3. Assume (2.1) for some λ > 8, (2.2) for some η > 12 and Em -δ 0 < ∞ for some δ > 0. Then for all t ∈ R, √ n 1 Π n Z n (ℓ n + s n t) -Φ(t)W n→∞ ----→ V(t) a.s., (2.3)
where

V(t) = - φ(t) V 1 (Eσ (2) 0 ) 1/2 + (Eσ (3) 0 ) (1 -t 2 ) φ(t) W 6(Eσ (2) 0 ) 3/2
. Theorem 2.4. Assume (2.1)for some λ > 16, (2.2) for some η > 16 and Em -δ 0 < ∞ for some δ > 0. Then for any bounded measurable set A ⊂ R with Lebesgue measure |A| > 0,

n √ 2πs n Π -1 n Z n (A + ℓ n ) -W A e -x 2 2s 2 n dx n→∞ ----→ µ(A) a.s., (2.4)
where

µ(A) = |A| 2Eσ (2) 0 -V 2 + 2 x A V 1 + |A| c(A) 8(Eσ (2) 0 ) 2 with x A = 1 |A| A xdx and c(A) = W E σ (4) 0 -3 σ (2) 0 2 + 4 (Eσ (3) 0 )(V 1 -x A W ) - 5(Eσ (3) 0 ) 2 3 Eσ (2) 0 W.
Remark 2.5. For a branching Wiener process, Theorems 2.3 and 2.4 improve Theorems 3.1 and 3.2 of Chen (2001, [START_REF] Chen | Exact convergence rates for the distribution of particles in branching random walks[END_REF]) by relaxing the second moment condition used by Chen to the moment condition of the form EX(ln + X) 1+λ < ∞ (cf. (2.1)). For a branching random walk with a constant or random environment, the second terms in V(•) and µ(•) are new: they did not appear in Chen's results [START_REF] Chen | Exact convergence rates for the distribution of particles in branching random walks[END_REF] for a branching Wiener process; the reason is that in the case of a Brownian motion, we have σ

(3) 0 = σ (4) 0 -3 σ (2) 0 2 = 0.
Remark 2.6. As will be seen in the proof, if we assume an exponential moment condition for the motion law, then the moment condition on the underlying branching mechanism can be weakened: in that case, we only need to assume that λ > 3/2 in Theorem 2.3 and λ > 4 in Theorem 2.4. In particular, for a branching Wiener process, Theorem 2.3 (resp. Theorem 2.4 ) is valid when (2.1) holds for some λ > 3/2 (resp. λ > 4).

Remark 2.7. In the deterministic case, Theorem 2.3 has been obtained by Kabluchko [40, Theorem 5 and Remark 2] under the second moment condition for the underlying branching mechanism.

Remark 2.8. When the Cramér condition P lim sup |t|→∞ E ξ e it L0 < 1 > 0 fails, the situation is different. Actually, while revising our manuscript we find that a lattice version (about a branching random walk on Z in a constant environment, for which the preceding condition fails) of Theorems 2.3 and 2.4 has been established very recently in [START_REF] Grübel | Asymptotic expansions for profiles of lattice branching random walks[END_REF].

For simplicity and without loss of generality, hereafter we always assume that l n = 0 (otherwise, we only need to replace L ui by L uil n ) and hence ℓ n = 0. In the following, we will write K ξ for a constant depending on the environment, whose value may vary from lines to lines.

Notation and Preliminary results

In this section, we introduce some notation and important lemmas which will be used in the sequel.

Notation

In addition to the σ-fields F n and D n , the following σ-fields will also be used:

I 0 = {∅, Ω}, I n = σ(ξ k , N u , L ui : k < n, i ≥ 1, |u| < n) for n ≥ 1.
For conditional probabilities and expectations, we write:

P ξ,n (•) = P ξ (•|D n ), E ξ,n (•) = E ξ (•|D n ); P n (•) = P(•|I n ), E n (•) = E(•|I n ); P ξ,Fn (•) = P ξ (•|F n ), E ξ,Fn (•) = E ξ (•|F n ).
As usual, we set

N * = {1, 2, 3, • • • } and denote by U = ∞ n=0 (N * ) n
the set of all finite sequences, where (N * ) 0 = {∅} contains the null sequence ∅.

For all u ∈ U , let T(u) be the shifted tree of T at u with defining elements {N uv }: we have 1)

∅ ∈ T(u), 2) vi ∈ T(u) ⇒ v ∈ T(u) and 3) if v ∈ T(u), then vi ∈ T(u) if and only if 1 ≤ i ≤ N uv . Define T n (u) = {v ∈ T(u) : |v| = n}. Then T = T(∅) and T n = T n (∅).
For every integer m ≥ 0, let H m be the Chebyshev-Hermite polynomial of degree m ( [START_REF] Petrov | Sums of independent random variables[END_REF]):

H m (x) = m! ⌊ m 2 ⌋ k=0 (-1) k x m-2k k!(m -2k)!2 k . (3.1)
The first few Chebyshev-Hermite polynomials relevant to us are:

H 0 (x) = 1, H 1 (x) = x, H 2 (x) = x 2 -1, H 3 (x) = x 3 -3x, H 4 (x) = x 4 -6x 2 + 3, H 5 (x) = x 5 -10x 3 + 15x, H 6 (x) = x 6 -15x 4 + 45x 2 -15, H 7 (x) = x 7 -21x 5 + 105x 3 -105x, H 8 (x) = x 8 -28x 6 + 210x 4 -420x 2 + 105.
It is known that ( [START_REF] Petrov | Sums of independent random variables[END_REF]) : for every integer m ≥ 0

Φ (m+1) (x) = d m+1 dx m+1 Φ(x) = (-1) m φ(x)H m (x).

Two preliminary lemmas

We first give an elementary lemma which will be often used in Section 4. (b) For each λ > 0, there exists a constant K λ > 0, such that

(ln + x) 1+λ ≤ K λ x, x > 0, (3.3)
(c) For each λ > 0, the function We next present the Edgeworth expansion for sums of independent random variables, that we shall need in Sections 5 and 6 to prove the main theorems. Let us recall the theorem used in this paper obtained by Bai and Zhao(1986, [START_REF] Bai | Edgeworth expansions of distribution functions of independent random variables[END_REF]), that generalizing the case for i.i.d random variables (cf. [33, P.159, Theorem 1]).

x(ln x) -1-λ is increasing for x > e 2λ . ( 3
Let {X j } be independent random variables, s atisfying for each j ≥ 1

EX j = 0, E|X j | k < ∞ with some integer k ≥ 3. (3.5)
We write B 2 n = n j=1 EX 2 j and only consider the nontrivial case B n > 0. Let γ νj be the ν-order cumulant of X j for each j ≥ 1. Write

λ ν,n = n (ν-2)/2 B -ν n n j=1 γ νj , ν = 3, 4 • • • , k; Q ν,n (x) = ′ (-1) ν+2s Φ (ν+2s) (x) ν m=1 1 k m ! λ m+2,n (m + 2)! km = -φ(x) ′ H ν+2s-1 (x) ν m=1 1 k m ! λ m+2,n (m + 2)! km ,
where the summation ′ is carried out over all nonnegative integer solutions (k 1 , . . . , k ν ) of the equations:

k 1 + • • • + k ν = s and k 1 + 2k 2 + • • • + νk ν = ν.
For 1 ≤ j ≤ n and x ∈ R, define

F n (x) = P B n -1 n j=1 X j ≤ x , v j (t) = Ee itXj ; Y nj = X j 1 {|Xj |≤Bn} , Z (x) nj = X j 1 {|Xj|≤Bn(1+|x|)} , W (x) nj = X j 1 {|Xj |>Bn(1+|x|)} .
The Edgeworth expansion theorem can be stated as follows.

Lemma 3.2 ([5]

). Let n ≥ 1 and X 1 , • • • , X n be a sequence of independent random variables satisfying (3.5) and B n > 0. Then for the integer k ≥ 3,

|F n (x) -Φ(x) - k-2 ν=1 Q νn (x)n -ν/2 | ≤ C(k) (1 + |x|) -k B -k n n j=1 E|W (x) nj | k + (1 + |x|) -k-1 B -k-1 n n j=1 E|Z (x) nj | k+1 + (1 + |x|) -k-1 n k(k+1)/2 sup |t|≥δn 1 n n j=1 |v j (t)| + 1 2n n , where δ n = 1 12 B 2 n ( n j=1 E|Y nj | 3 ) -1 , C(k) > 0 is a constant depending only on k.
4 Convergence of the martingales {(N 1,n , D n )} and {(N 2,n , D n )}

Now we can proceed to prove the convergence of the two martingales defined in Section 2.

Convergence of the martingale

{(N 1,n , D n )}
Proof of Proposition 2.1 . The fact that {(N 1,n , D n )} is a martingale can be easily shown: it suffices to notice that

E ξ,n N 1,n+1 = E ξ,n 1 Π n+1 u∈Tn+1 S u = 1 Π n+1 E ξ,n u∈Tn Nu i=1 (S u + L ui ) = 1 Π n+1 u∈Tn E ξ,n Nu i=1 (S u + L ui ) = 1 Π n+1 u∈Tn m n S u = N 1,n .
Observe that

N 1,n+1 -N 1,n = 1 Π n u∈Tn S u (N u /m n -1) + 1 Π n u∈Tn ( 1 m n Nu i=1 L ui ) =: I 1,n + I 2,n . (4.1)
We shall prove the convergence of the martingale by showing that both of the series

∞ n=1 I 1,n and ∞ n=1 I 2,n converge a.s. (4.2) Define I ′ 1,n = 1 Π n u∈Tn S u ( N u m n -1)1 {Nu/mn≤Πn} , (4.3)
I ′ 2,n = 1 Π n u∈Tn Y u 1 {|Yu|≤Πn} with Y u = 1 m |u| Nu i=1 L ui . (4.4) Since lim n→∞ 1 n n j=1 E ξ | L j | q = E| L 1 | q < ∞, q = 1, 2,
there exists a constant K ξ < ∞ depending only on ξ such that for n ≥ 1 and |u| = n,

E ξ | L n | ≤ K ξ n, E ξ |S u | ≤ n j=1 E ξ | L j | ≤ K ξ n, E ξ |S u | 2 = n j=1 E ξ | L j | 2 ≤ K ξ n. (4.5)
To prove the convergence a.s. of the series ∞ n=1 I q,n (q = 1, 2), we only need to show that the following series converges a.s.:

∞ n=1 (I q,n -I ′ q,n ), ∞ n=1 (I ′ q,n -E ξ,n I ′ q,n ), and ∞ n=1 E ξ,n I ′ q,n .
First for q = 1, we see that

E ξ |I 1,n -I ′ 1,n | = E ξ | 1 Π n u∈Tn S u ( N u m n -1)1 {Nu/mn>Πn} | ≤ E ξ 1 Π n u∈Tn E ξ |S u |E ξ ( N u m n + 1)1 {Nu/mn>Πn} ≤ K ξ nE ξ ( N n m n + 1)1 { Nn/mn>Πn} ≤ K ξ n (ln(Π n + 1)) 1+λ E ξ N n m n + 1 ln 1 + N n m n 1+λ ≤ K ξ n -λ E ξ N n m n + 1 ln 1 + N n m n 1+λ .
Observe that for λ > 1,

E ∞ n=1 1 n λ E ξ N n m n (ln + N n ) 1+λ + (ln -m n ) 1+λ = ∞ n=1 1 n λ E N 0 m 0 (ln + N 0 ) 1+λ + E(ln -m 0 ) 1+λ < ∞, which implies that ∞ n=1 1 n λ E ξ N n m n (ln + N n ) 1+λ + (ln -m n ) 1+λ < ∞ a.s. (4.6) Hence E ξ | ∞ n=1 (I 1,n -I ′ 1,n )| ≤ ∞ n=1 E ξ |I 1,n -I ′ 1,n | < ∞, E ξ | ∞ n=1 E ξ,n I ′ 1,n | = E ξ | ∞ n=1 E ξ,n (I 1,n -I ′ 1,n )| ≤ ∞ n=1 E ξ |I 1,n -I ′ 1,n | < ∞. It follows that the series ∞ n=1 (I 1,n -I ′ 1,n ) and ∞ n=1 E ξ,n I ′ 1,n converge a.s. Observe that n k=1 (I ′ 1,k -E ξ,D k I ′ 1,k
) is a martingale w.r.t. {D n+1 }. By the a.s. convergence of an L 2 bounded martingale (see e.g. [15, P. 251, Ex. 4.9]), we prove the convergence a.s. of the series

∞ n=1 (I ′ 1,n -E ξ,n I ′ 1,n ) by showing that of the series ∞ n=1 E ξ (I ′ 1,n -E ξ,n I ′ 1,n ) 2 .
It is immediate from the following:

E ξ (I ′ 1,n -E ξ,n I ′ 1,n ) 2 = E ξ 1 Π n u∈Tn S u ( N u m n -1)1 { Nu mn ≤Πn} -E ξ,n ( N u m n -1)1 { Nu mn ≤Πn} 2 = E ξ 1 Π 2 n u∈Tn S 2 u E ξ,n ( N u m n -1)1 { Nu mn ≤Πn} -E ξ,n ( N u m n -1)1 { Nu mn ≤Πn} 2 ≤ E ξ 1 Π 2 n u∈Tn S 2 u E ξ N u m n -1 2 1 { Nu mn ≤Πn} ≤ E ξ 1 Π 2 n u∈Tn E ξ S 2 u E ξ N u m n 2 1 { Nu mn ≤Πn} + 3 = s 2 n Π n E ξ N n m n 2 1 { Nn mn ≤Πn} + 3s 2 n Π n = s 2 n Π n E ξ N n m n 2 1 { Nn mn ≤min(e 2λ ,Πn)} + 1 {e 2λ < Nn mn ≤Πn} + 3s 2 n Π n ≤ (3.4) (3 + e 4λ ) s 2 n Π n + s 2 n Π n E ξ N n m n 2 1 + N n m n ln -(1+λ) 1 + N n m n -1 (Π n + 1) ln -1-λ 1 + Π n ≤ (3 + e 4λ ) s 2 n Π n + K ξ s 2 n ln 1+λ 1 + Π n E ξ N n m n ln 1+λ 1 + N n m n ≤ K ξ n Π n + K ξ 1 n λ E ξ N n m n ln 1+λ 1 + N n m n .
Combining the above results, we see that the series I 1,n converges a.s.

Next we turn to the proof of the convergence a.s. of the series I 2,n .

To begin with, we prove that

E ξ |Y u |(ln + |Y u |) 1+λ ≤ K ξ n + K ξ n(ln -m n ) 1+λ . (4.7)
This follows from the fact:

E ξ |Y u |(ln + |Y u |) 1+λ ≤ E ξ 1 m n Nu i=1 L ui ln + Nu i=1 L ui + ln -m n 1+λ ≤ E ξ 1 m n Nu i=1 L ui 2 λ ln + Nu i=1 L ui 1+λ + (ln -m n ) 1+λ ≤ (3.3) K λ 1 m n E ξ Nu i=1 L ui 2 + 2 λ (ln -m n ) 1+λ 1 m n E ξ Nu i=1 L ui ≤ K λ 1 m n E ξ Nu i=1 E ξ |L ui | 2 + 2 λ (ln -m n ) 1+λ 1 m n E ξ Nu i=1 E ξ L ui ≤ K ξ n + K ξ n(ln -m n ) 1+λ .
Observe that

E ξ |I 2,n -I ′ 2,n | = E ξ | 1 Π n u∈Tn Y u 1 {|Yu|>Πn} | ≤ E ξ 1 Π n u∈Tn |Y u |1 {|Yu|>Πn} ≤ E ξ 1 Π n (ln Π n ) 1+λ u∈Tn E ξ |Y u |(ln + |Y u |) 1+λ ≤ K ξ n + K ξ n(ln -m n ) 1+λ (ln Π n ) 1+λ ≤ K ξ n λ (1 + (ln -m n ) 1+λ )
By (4.6), the series n -λ (1 + (ln -m n ) 1+λ ) converges a.s. Thus

E ξ | ∞ n=1 (I 2,n -I ′ 2,n )| ≤ ∞ n=1 E ξ |I 2,n -I ′ 2,n | < ∞, E ξ | ∞ n=1 E ξ,n I ′ 2,n | = E ξ | ∞ n=1 E ξ,n (I 2,n -I ′ 2,n )| ≤ ∞ n=1 E ξ |I 2,n -I ′ 2,n | < ∞.
This implies the convergence a.s. of the series

∞ n=1 (I 2,n -I ′ 2,n ) and ∞ n=1 E ξ,n I ′ 2,n . To prove the convergence a.s. of the series ∞ n=1 (I ′ 2,n -E ξ,n I ′ 2,n
), we only need to show the convergence of the series:

∞ n=1 E ξ (I ′ 2,n -E ξ,n I ′ 2,n ) 2
. This is implied by the following observation:

E ξ (I ′ 2,n -E ξ,n I ′ 2,n ) 2 = E ξ 1 Π 2 n u∈Tn E ξ |Y u |1 {|Yu|≤Πn} -E ξ |Y u |1 {|Yu|≤Πn} 2 ≤ E ξ 1 Π 2 n u∈Tn E ξ |Y u | 2 1 {|Yu|≤Πn} ≤ E ξ 1 Π 2 n u∈Tn E ξ |Y u | 2 1 {|Yu|≤min(e 2λ ,Πn)} + |Y u | 2 1 {e 2λ <|Yu|≤Πn} ≤ (3.4) e 4λ Π n + E ξ 1 Π 2 n u∈Tn E ξ |Y u | 2 Π n (ln Π n ) -1-λ (|Y u |(ln + |Y u |) -1-λ ) -1 ≤ e 4λ Π n + E ξ 1 Π n (ln Π n ) 1+λ u∈Tn E ξ |Y u |(ln + |Y u |) 1+λ ≤ (4.7) e 4λ Π n + K ξ n + K ξ n(ln -m n ) 1+λ (ln Π n ) 1+λ ≤ e 4λ Π n + K ξ n λ (1 + (ln -m n ) 1+λ ).
Combining the above results, we see that the series I 2,n converges a.s. Therefore we have proved (4.2) and the martingale {N 1,n } converges a.s. to

V 1 := ∞ n=1 (N 1,n+1 -N 1,n ) + N 1,1 .

Convergence of the martingale

{(N 2,n , D n )} Proof of Proposition 2.2.
To see that {(N 2,n , D n )} is a martingale, it suffices to notice that (remind that we have assumed

ℓ n = 0) E ξ,n N 2,n+1 = E ξ,n 1 Π n+1 u∈Tn+1 (S 2 u -s 2 n+1 ) = 1 Π n+1 u∈Tn E ξ,n Nu i=1 (S u + L ui ) 2 -s 2 n+1 = 1 Π n+1 u∈Tn E ξ,n Nu i=1 (S 2 u + 2S u L ui + L 2 ui -s 2 n+1 ) = 1 Π n+1 u∈Tn E ξ,n Nu i=1 E ξ,n (S 2 u + 2S u L ui + L 2 ui -s 2 n+1 ) N u = 1 Π n+1 u∈Tn m n (S 2 u + σ (2) n -s 2 n+1 ) = 1 Π n u∈Tn (S 2 u -s 2 n ) = N 2,n .
As in the proof of Proposition 2.1, we will prove the convergence of the martingale

{(N 2,n , D n )} by showing that ∞ n=1 (N 2,n+1 -N 2,n ) converges a.s.
We start by giving some notation. For n ≥ 1 and |u| = n, set

J 1,n := 1 Π n u∈Tn (S 2 u -s 2 n )( N u m n -1), J ′ 1,n := 1 Π n u∈Tn (S 2 u -s 2 n )( N u m n -1)1 { Nu mn ≤Πn} ; (4.8)
J 2,n := 1 Π n u∈Tn Q u , J ′ 2,n := 1 Π n u∈Tn Q u 1 {|Qu|≤Πn} , with Q u = 1 m n Nu i=1 (L 2 ui -σ (2) n ); (4.9)
J 3,n := 1 Π n u∈Tn S u Y u , J ′ 3,n := 1 Π n u∈Tn S u Y u 1 {|Yu|≤Πn} , with Y u = 1 m n Nu i=1 L ui . (4.10)
Then we see that

N 2,n+1 -N 2,n = J 1,n + J 2,n + 2J 3,n .
Thus the convergence a.s. of the martingale {N 2,n } will follow from the convergence a.s. of the series J q,n (q = 1, 2, 3). Using the above truncation and following the same procedure as in the proofs of Proposition 2.1 we can prove the convergence a.s. of the series J q,n (q = 1, 2, 3). We omit the details and only stress the following estimate:

E ξ |Q u |(ln + |Q u |) 1+λ = E ξ 1 m n Nu i=1 (L 2 ui -σ (2) n ) ln + 1 m n Nu i=1 (L 2 ui -σ (2) n ) 1+λ ≤ E ξ 1 m n Nu i=1 (L 2 ui -σ (2) n ) ln + Nu i=1 (L 2 ui -σ (2) n ) + ln -m n 1+λ ≤ 2 λ E ξ 1 m n Nu i=1 (L 2 ui -σ (2) n ) (ln + Nu i=1 (L 2 ui -σ (2) n ) ) 1+λ + (ln -m n ) 1+λ ≤ K λ E ξ 1 m n Nu i=1 (L 2 ui -σ (2) n ) 2 + K ξ nE ξ (ln -m n ) 1+λ ≤ K ξ n + K ξ nE ξ (ln -m n ) 1+λ .
So the martingale N 2,n converges a.s. to the limit

V 2 := ∞ n=1 (N 2,n+1 -N 2,n ) + N 2,1 .
5 Proof of Theorem 2.3

A key decomposition

For u ∈ (N * ) k (k ≥ 0) and n ≥ 1, write for B ⊂ R,

Z n (u, B) = v∈Tn(u) 1 B (S uv -S u ).
Then we see that |X n,u | ≤ W n-kn (u) + 1.

To prove Lemma 5.1, we will use the extended Borel-Cantelli Lemma. We can obtain the required result once we prove that ∀ε > 0,

∞ n=1 P kn (| √ nA n | > 2ε) < ∞. (5.6)
Notice that

P kn (|A n | > 2 ε √ n ) ≤ P kn (A n = Ān ) + P kn (| Ān -E ξ,kn Ān | > ε √ n ) + P kn (|E ξ,kn Ān | > ε √ n ).
We will proceed the proof in 3 steps.

Step 1 We first prove that

∞ n=1 P kn (A n = A n ) < ∞. (5.7)
To this end, define

W * = sup n W n ,
and we need the following result :

Lemma 5.4. ([28, Th. 1.2]) Assume (2.1) for some λ > 0 and Em -δ 0 < ∞ for some δ > 0. Then

E(W * + 1)(ln(W * + 1)) λ < ∞. (5.8)
We observe that

P kn (A n = A n ) ≤ u∈T kn P kn (X n,u = X n,u ) = u∈T kn P kn (|X n,u | ≥ Π kn ) ≤ u∈T kn P kn (W n-kn (u) + 1 ≥ Π kn ) = W kn r n P(W n-kn + 1 ≥ r n ) rn=Π kn ≤ W kn E (W n-kn + 1)1 {W n-kn +1≥rn} rn=Π kn ≤ W kn E (W * + 1)1 {W * +1≥rn} rn=Π kn ≤ W * (ln Π kn ) -λ E(W * + 1)(ln(W * + 1)) λ ≤ K ξ W * n -λβ E(W * + 1)(ln(W * + 1)) λ ,
where the last inequality holds since

1 n ln Π n → E ln m 0 > 0 a.s. , (5.9)
and k n ∼ n β . By the choice of β and Lemma 5.4, we obtain (5.7).

Step 2. We next prove that ∀ε > 0,

∞ n=1 P kn (|A n -E ξ,kn A n | > ε √ n ) < ∞. (5.10)
Take a constant b ∈ (1, e E ln m0 ). Observe that ∀u ∈ T kn , n ≥ 1, Then we have that

E kn X2 n,u = ∞ 0 2xP kn (| Xn,u | > x)dx = 2 ∞ 0 xP kn (|X n,u |1 {|Xn,u|<Π kn } > x)dx ≤ 2 Π kn 0 xP kn (|W n-kn (u) + 1| > x)dx = 2 Π kn 0 xP(|W n-kn + 1| > x)dx ≤ 2 Π kn 0 xP(W * + 1 > x)dx ≤ 2 Π kn e ( ln
∞ n=1 P kn (|A n -E ξ,kn A n | > ε √ n ) = ∞ n=1 E kn P ξ,kn (|A n -E ξ,kn A n | > ε √ n ) ≤ ε -2 ∞ n=1 nE kn   Π -2 kn u∈T kn E ξ,kn X 2 n,u   = ε -2 ∞ n=1 n   Π -2 kn u∈T kn E kn X 2 n,u   ≤ ε -2 ∞ n=1 nW kn Π kn 2E(W * + 1)(ln(W * + 1) λ )(b kn + (Π kn -b kn )(k n ln b) -λ ) + 9 ≤ 2ε -2 W * E(W * + 1)(ln(W * + 1) λ ) ∞ n=1 n Π kn b kn + ∞ n=1 n(k n ln b) -λ + 9ε -2 W * ∞ n=1 n Π kn .
By (5.9) and λβ > 2, the three series in the last expression above converge under our hypothesis and hence (5.10) is proved.

Step 3. Observe

P kn |E ξ,kn Ān | > ε √ n ≤ √ n ε E kn |E ξ,kn Ān | = √ n ε E kn 1 Π kn u∈T kn E ξ,kn Xn,u = √ n ε E kn 1 Π kn u∈T kn (-E ξ,kn X n,u 1 {|Xn,u|≥Π kn } ) ≤ √ n ε 1 Π kn u∈T kn E kn (W n-kn (u) + 1)1 {W n-kn (u)+1≥Π kn } = √ nW kn ε E(W n-kn + 1)1 {W n-kn +1≥rn} rn=Π kn ≤ W * ε √ n E(W * + 1)1 {W * +1≥rn} rn=Π kn ≤ W * ε √ n (ln Π kn ) λ E(W * + 1) ln λ (W * + 1) ≤ W * ε K ξ n 1 2 -λβ E(W * + 1) ln λ (W * + 1).
Then by (5.9) and λβ > 2, it follows that

∞ n=1 P kn |E ξ,kn Ān | > ε √ n < ∞.
Combining Steps 1-3, we obtain (5.6). Hence the lemma is proved.

Proof of Lemma 5.2 . For ease of notation, set

D 1 (t) = (1 -t 2 )φ(t), κ 1,n = s (3) n -s (3) kn 6(s 2 n -s 2 kn ) 3/2 .
Observe that

B n = B n1 + B n2 + B n3 + B n4 , (5.11)
where

B n1 = 1 Π kn u∈T kn E ξ,kn W n-kn (u, s n t -S u ) -Φ s n t -S u (s 2 n -s 2 kn ) 1/2 -κ 1,n D 1 s n t -S u (s 2 n -s 2 kn ) 1/2 ; B n2 = 1 Π kn u∈T kn Φ s n t -S u (s 2 n -s 2 kn ) 1/2 -Φ(t) ; B n3 = κ 1,n 1 Π kn u∈T kn D 1 s n t -S u (s 2 n -s 2 kn ) 1/2 -D 1 (t) ; B n4 = κ 1,n D 1 (t)W kn .
Then the lemma will be proved once we show that √ nB n1 n→∞ ----→ 0;

(5.12)

√ nB n2 n→∞ ----→ -(Eσ (2) 0 ) -1 2 φ(t)V 1 ; (5.13) √ nB n3 n→∞ ----→ 0; (5.14) √ nB n4 n→∞ ----→ 1 6 Eσ (3) 0 (Eσ (2)
0 ) -3 2 D 1 (t)W. (5.15)
We will prove these results subsequently. We first prove (5.12). The proof will mainly be based on the following result about asymptotic expansion of the distribution of the sum of independent random variables: Proposition 5.5. Under the hypothesis of Theorem 2.3, for a.e. ξ,

ǫ n = n 1/2 sup x∈R P ξ n-1 k=kn L k (s 2 n -s 2 kn ) 1/2 ≤ x -Φ(x) -κ 1,n D 1 (x) n→∞ ----→ 0. Proof. Let X k = 0 for 0 ≤ k ≤ k n -1 and X k = L k for k n ≤ k ≤ n -1.
Then the random variables {X k } are independent under P ξ . Denote by v k (•) the characteristic function of

X k : v k (t) := E ξ e itX k .
Using the Markov inequality and Lemma 3.2, we obtain the following result:

sup x∈R P ξ n-1 k=kn L k (s 2 n -s 2 kn ) 1/2 ≤ x -Φ(x) -κ 1,n D 1 (x) ≤K ξ    (s 2 n -s 2 kn ) -2 n-1 j=kn E ξ | L j | 4 + n 6   sup |t|>T 1 n k n + n-1 j=kn |v j (t)| + 1 2n   n    .
By our conditions on the environment, we know that Then Ec 0 < 1. By the Birkhoff ergodic theorem, we have

lim n→∞ n(s 2 n -s 2 kn ) -2 n-1 j=kn E ξ | L k | 4 = E| L 0 | 4 /(Eσ (2) 0 ) 2 . ( 5
sup |t|>T 1 n n-1 j=kn |v j (t)| ≤ 1 n n-1 j=1 c j → Ec 0 < 1.
Then for n large enough,   sup

|t|>T 1 n k n + n-1 j=kn |v j (t)| + 1 2n   n = o(n -m ), ∀m > 0.
(5.17)

From (5.16) and (5.17), we get the conclusion of the proposition.

From Proposition 5.5, it is easy to see that

√ n|B n1 | ≤ W kn ǫ n n→∞ ----→ 0.
Hence (5.12) is proved. We next prove (5.13). Observe that

B n2 = B n21 + B n22 + B n23 + B n24 + B n25 , with B n21 = 1 Π kn u∈T kn Φ s n t -S u (s 2 n -s 2 kn ) 1/2 -Φ(t) -φ(t) s n t -S u (s 2 n -s 2 kn ) 1/2 -t 1 {|Su|≤kn} , B n22 = 1 Π kn u∈T kn Φ s n t -S u (s 2 n -s 2 kn ) 1/2 -Φ(t) 1 {|Su|>kn} , B n23 = - 1 Π kn u∈T kn s n t -S u (s 2 n -s 2 kn ) 1/2 -t φ(t)1 {|Su|>kn} , B n24 = 1 (s 2 n -s 2 kn ) 1/2 s n -(s 2 n -s 2 kn ) 1/2 W kn φ(t)t, B n25 = - 1 (s 2 n -s 2 kn ) 1/2 φ(t)N 1,kn .
By Taylor's formula and the choice of β and k n , we get

ǫ n = √ n sup |y|≤kn Φ s n t -y (s 2 n -s 2 kn ) 1/2 -Φ(t) -φ(t) s n t -y (s 2 n -s 2 kn ) 1/2 -t ≤ √ n sup |y|≤kn s n t -y (s 2 n -s 2 kn ) 1/2 -t 2 n→∞ ----→ 0. Thus | √ nB n21 | ≤ W kn ǫ n n→∞ ----→ 0. (5.18)
We continue to prove that .19) This will follow from the facts: .20) In order to prove (5.20), we firstly observe that

√ nB n22 n→∞ ----→ 0; √ nB n23 n→∞ ----→ 0. ( 5
1 Π kn u∈T kn |S u |1 {|Su|>kn} n→∞ ----→ 0 a.s.; √ n 1 Π kn u∈T kn 1 {Su|>kn} n→∞ ----→ 0 a.s. ( 5
E   ∞ n=1 1 Π kn u∈T kn |S u |1 {|Su|>kn}   = ∞ n=1 E| S kn |1 {| S kn |>kn} ≤ ∞ n=1 k 1-η n E| S kn | η ≤ ∞ n=1 k -η 2 n kn-1 j=0 E| L j | η = ∞ n=1 k 1-η 2 n E| L 0 | η , E   ∞ n=1 √ n 1 Π kn u∈T kn 1 {|Su|>kn}   = ∞ n=1 √ nE1 {| S kn |>kn} ≤ ∞ n=1 √ nk -η n E| S kn | η ≤ ∞ n=1 √ nk -η 2 -1 n kn-1 j=0 E| L j | η = ∞ n=1 n 1 2 k -η 2 n E| L 0 | η .
The assumptions on β, k n and η ensure that the series in the right hand side of the above two expressions converge. Hence

∞ n=1 1 Π kn u∈T kn |S u |1 {|Su|>kn} < ∞, ∞ n=1 √ n 1 Π kn u∈T kn 1 {|Su|>kn} < ∞ a.
whence by the choice of β < 1/4 and the conditions on the environment, The first term in the last expression above tends to 0 a.s. by (5.20), and the second one tends to 0 a.s. because the martingale {W n } converges and

√ nB n24 = √ n (s 2 n -s 2 kn ) 1/2 s 2 kn s n + (s 2 n -s 2 kn ) 1/2 W kn φ(t)t n→∞ ----→ 0. ( 5
sup |y|≤kn D 1 s n t -y (s 2 n -s 2 kn ) 1/2 -D 1 (t) n→∞ ----→ 0.
Combining the above results, we obtain (5.14). It remains to prove (5.15), which is immediate from (5.24) and the fact W n n→∞ ----→ W . So Lemma 5.2 has been proved. The required proposition concludes from (6.13) and (5.17

Lemma 3 . 1 .

 31 (a) For x, y ≥ 0, ln + (x + y) ≤ 1 + ln + x + ln + y, ln(1 + x) ≤ 1 + ln + x. (3.2)

. 4 )

 4 Proof. Part (a) holds since ln + (x + y) ≤ ln + (2 max{x, y}) ≤ 1 + ln + x + ln + y. Parts (b) and (c) can be verified easily.

9 ≤ 9 ≤

 99 x) -λ E(W * + 1)(ln(W * + 1)) λ dx + 2E(W * + 1)(ln(W * + 1)) λ b kn e (ln x) -λ dx + Π kn b kn (ln x) -λ dx + 2E(W * + 1)(ln(W * + 1)) λ (b kn + (Π knb kn )(k n ln b) -λ ) + 9.

Proposition 6 . 4 .-s 2 kn ) 1 / 2 ≤ 3 ν=1κ-s 2 kn ) 1 / 2 ≤ 3 ν=1κ

 6421232123 Under the hypothesis of Theorem 2.4, for a.e. ξ,ǫ n = n 3/2 sup x∈R x -Φ(x)ν,n D ν (x) -R n (x) n→∞ ----→ 0. Proof. Let X k = 0 for 0 ≤ k ≤ k n -1 and X k = L k for k n ≤ k ≤ n -1.Then the random variables {X k } are independent under P ξ . By Markov's inequality and Lemma 3.2 we obtain the following result:x -Φ(x)ν,n D ν (x) -R n (x)By our conditions on the environment, we know thatlim n→∞ n 2 (s 2 ns 2 kn ) -3 n-1 j=kn E ξ | L k | 6 = E| L 0 | 6 /(Eσ

  So there exists a constant c n ≤ 1 depending on ξ n such that sup

.16) By (2.2), L n satisfies P lim sup |t|→∞ |v n (t)| < 1 > 0. |t|>T |v n (t)| ≤ c n and P(c n < 1) > 0.

  From (5.18),(5.19),(5.22) and (5.23), we derive (5.13). Now we turn to the proof of(5.14). According to the hypothesis of Theorem 2.3, it follows from the Birkhoff ergodic theorem that

							.22)
	Due to Proposition 2.1 and (5.21), we conclude that
				√ nB n25	n→∞ ----→ -(Eσ 0 ) -1 (2) 2 φ(t)V 1 a.s.	(5.23)
					lim n→∞	√	nκ 1,n =	1 6	(Eσ 0 ) -3/2 Eσ (2) 0 . (3)	(5.24)
	Notice that
		1 Π kn u∈T kn	D 1	s n t -S u (s 2 n -s 2 kn ) 1/2 -D 1 (t)
	≤	2 Π kn u∈T kn	1 {|Su|>kn} +	1 Π kn u∈T kn	D 1	s n t -S u (s 2 n -s 2

kn ) 1/2 -D 1 (t) 1 {|Su|≤kn} .

  ). and (6.6) is proved.Next we turn to the proof of (6.7). Using Taylor's expansion and the boundedness of the set A, together with the choice of β and k n , we get that uniformly for all |y| ≤ k n and x ∈ A as n → ∞. By the same arguments as in the proof of (5.20), we can show that for η > 16, with β, k n chosen above,

	Using Proposition 6.4, we deduce that
							|nΛ 2,n1 | ≤	√ 2πs n n -1 2 W kn ǫ n	n→∞ ----→ 0,
	s n n -s 2 (s 2 kn ) 1/2 exp{-	(x -y) 2 2(s 2 n -s 2 kn )	} -exp{-	x 2 2s 2 n	} =	1 n -s 2 2(s 2 kn )	s 2 kn -y 2 + 2xy + o(1) ,
	nΠ -1 kn	u∈T kn	1 {|Su|>kn}	n→∞ ----→ 0 and	Π -1 kn	u∈T kn	S 2 u 1 {|Su|>kn}	n→∞ ----→ 0 a.s.	(6.14)
	Therefore as n tends to infinity, we have a.s.
				nΛ 2,n2 = n	1 n -s 2 2(s 2 kn )	-|A|Π -1 kn	u∈T kn	(S 2 u -s 2 kn )1 {|Su|≤kn}
						+ 2	A	xdxΠ -1 kn	u∈T kn	S u 1 {|Su|≤kn} + o(1)
				= = (2Eσ 2(s 2 n -s 2 n kn ) (2) 0 ) -1 (-V 2 + 2x A V 1 )|A| + o(1), -N 2,kn |A| + 2|A|x A N 1,kn + o(1)
	which proves (6.7).
	To prove (6.8), we observe that
	Λ 2,n3 =	κ 1,n s n n -s 2 (s 2 kn ) 1/2 Π -1 kn	u∈T kn	1 {|Su|≤kn}	A	(x -S u ) 3 (s 2 n -s 2 kn ) 3/2 -	3(x -S u ) (s 2 n -s 2 kn ) 1/2 e	-(x-Su) 2 2(s 2 n -s 2 kn

) dx = Λ 2,n31 + Λ 2,n32 + Λ 2,n33 + Λ 2,n34 ,

Acknowledgments

The authors are grateful to the reviewers for valuable remarks and comments which lead to a significant improvement of our original manuscript. The work has benefited from a visit of Q. Liu to the School of Mathematical Sciences, Beijing Normal University, and a visit of Z. Gao to Laboratoire de Mathématiques de Bretagne Atlantique, Université de Bretagne-Sud. The hospitality and support of both universities have been well appreciated.

* The project is partially supported by the National Natural Science Foundation of China (NSFC, Grants No. 11101039, No. 11171044, No. 11271045, and No. 11401590), by a cooperation program between NSFC and CNRS of France (Grant No. 11311130103), by the Fundamental Research Funds for the Central Universities (2013YB52) and by Hunan Provincial Natural Science Foundation of China (Grant No. 11JJ2001).

It can be easily seen that the law of Z n (u, B) under P ξ is the same as that of Z n (B) under P θ k ξ . Define

By definition, we have

The following decomposition will play a key role in our approach: for k ≤ n,

(5.1)

Remark that by our definition, for u ∈ T k ,

represents number of the descendants of u at time n situated in B.

For each n, we choose an integer k n < n as follows. Let β be a real number such that max { 2 λ , 3 η } < β < 1 4 and set k n = ⌊n β ⌋, the integral part of n β . Then on the basis of (5.1), the following decomposition will hold:

where

Here we remind that the random variables W n-kn (u, s n t -S u ) are independent of each other under the conditional probability P ξ,kn .

Proof of Theorem 2.3

First, observe that the condition Now we go to prove the lemmas subsequently.

Proof of Lemma 5.1. For ease of notation, we define for |u| = k n ,

Proof of Lemma 5.3. This lemma follows from the following result given in [START_REF] Huang | Convergence rates for a branching process in a random environment[END_REF].

Proposition 5.6 ( [24]

). Assume the condition (2.1). Then

By the choice of β and k n , we see that

Now Theorem 2.3 follows from the decomposition (5.2) and Lemmas 5.1 -5.3.

6 Proof of Theorem 2.4

We will follow the similar procedure as in the proof of Theorem 2.3. We remind that λ, η > 16 in the current setting. Hereafter we will choose max{ 4 λ , 4 η } < β < 1 4 and let k n = ⌊n β ⌋ (the integral part of n β ).

By (5.1), we have

with

On basis of this decomposition, we shall divide the proof of Theorem 2.4 into the following lemmas.

Under the hypothesis of Theorem 2.4, a.s.

Now we go to prove the lemmas subsequently.

Proof of Lemma 6.1. The proof of Lemma 6.1 follows the same procedure as that of Lemma 5.1 with minor changes in scaling. We omit the details.

Proof of Lemma 6.2. We start the proof by introducing some notation: set

Define for x ∈ R,

ns

where H m are Chebyshev-Hermite polynomials defined in (3.1). We decompose Λ 2,n into 7 terms:

where

The lemma will follow once we prove that a.s.

)

)

)

)

)

)

The proof of (6.6) is based on the following result on the asymptotic expansion of the distribution of the sum of independent random variables: with

It is clear that

ns Combining (6.14),(6.15), (6.16) and (6.17), we deduce (6.9) and (6.10). By the Birkhoff ergodic theorem and the definition of H m (x) and φ(x), we see that

whence (6.11) follows.

Finally because |Λ 2,n7 | is bounded by K ξ • Π -1 kn u∈T kn 1 {|Su|>kn} , (6.14) implies (6.12). So the required result (6.3) follows from (6.6) -(6.12).

Proof of Lemma 6.3. By Proposition 5.6 , under our assumption, we have

By the choice of β and k n , we see that