Jérôme Dedecker

Xiequan Fan

Haijuan Hu

Florence Merlevède

Rates of convergence in the central limit theorem for the elephant random walk with random step sizes

Keywords: Elephant random walk, law of the iterated logarithm, normal approximations, Wassertein's distance, Berry-Esseen bounds, central limit theorem Mathematics Subject Classification (2010): 60G42, 60F05, 60E15, 82C41

In this paper, we consider a generalization of the elephant random walk model. Compared to the usual elephant random walk, an interesting feature of this model is that the step sizes form a sequence of positive independent and identically distributed random variables instead of a fixed constant. For this model, we establish the law of the iterated logarithm, the central limit theorem, and we obtain rates of convergence in the central limit theorem with respect to the Kologmorov, Zolotarev and Wasserstein distances. We emphasize that, even in case of the usual elephant random walk, our results concerning the rates of convergence in the central limit theorem are new.

Introduction

The elephant random walk (ERW) is a type of one-dimensional random walk on integers, which has a complete memory of its whole history. The ERW was introduced in 2004 by Schütz and Trimper [24] in order to study the memory effects of a non-Markovian random walk. The model has a link to a famous saying that elephants can always remember where they have been.

The ERW can be defined as follows. It starts at time n = 0, with position T 0 = 0. At time n = 1, the elephant moves to 1 with probability q and to -1 with probability 1 -q, where q ∈ (0, 1]. So the position of the elephant at time n = 1 is given by T 1 = X 1 , with X 1 a Rademacher R(q) random variable. At time n + 1, n ≥ 1, the step X n+1 is determined stochastically by the following rule. Let n be an integer which is chosen from the set {1, 2, . . . , n} uniformly at random. If X n = 1, then X n+1 = 1 with probability p -1 with probability 1 -p.

If X n = -1, then X n+1 = 1
with probability 1 -p -1 with probability p.

Equivalently, X n+1 is determined stochastically by the following rule:

X n+1 =
X n with probability p -X n with probability 1 -p.

The last equality suggests that at time n + 1, we reinforce X n with probability p and reduce X n with probability 1 -p. Thus, for n ≥ 2, the position of the elephant at time n is

T n = n i=1 X i , where X n = α n X βn , (1.1)
with α n has a Rademacher distribution R(p), p ∈ (0, 1], and β n is uniformly distributed over the integers {1, 2, . . . , n -1}. Moreover, α n is independent of (X i ) 1≤i≤n , and the random variables (β i ) i≥1 are independent. Here p is called the memory parameter. The ERW is respectively called diffusive, critical and superdiffusive according to p ∈ (0, 3/4), p = 3/4 and p ∈ (3/4, 1]. The description of the asymptotic behavior of the ERW has motivated many interesting works. By showing the connection of the ERW model with Polya-type urns, Baur and Bertoin obtained the functional form of the central limit theorem. Coletti, Gava and Schütz [START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF][START_REF] Coletti | A strong invariance principle for the elephant random walk[END_REF] proved the central limit theorem (CLT) and a strong invariance principle for p ∈ (0, 3/4] and a law of large numbers for p ∈ (0, 1). Moreover, they also showed that if p ∈ (3/4, 1], then the ERW converges to a non-degenerate random variable which is not normal. Vázquez Guevara [START_REF] Vázquez Guevara | On the almost sure central limit theorem for the elephant random walk[END_REF] gave the almost sure CLT. Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] recovered the CLT via a martingale method. Bertoin [START_REF] Bertoin | Counting the zeros of an elephant random walk[END_REF] studied how memory impacts passages at the origin for the ERW. Recently, a number of variations for the ERW has been introduced. For instance, Bercu and Lucile [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] introduced a multi-dimensional type ERW, and established the multivariate CLT. Gut and Stadtmüller [START_REF] Gut | Variations of the elephant random walk[END_REF] investigated the case when the elephant remembers only a finite part of the first or last steps. Laulin [START_REF] Laulin | New insights on the reinforced elephant random walk using a martingale approach[END_REF] studied the asymptotic analysis for the reinforced ERW model. Recently, Bercu [START_REF] Bercu | On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution[END_REF] considered the ERW with stops playing hide and seek with the Mittag-Leffler distribution.

In this paper, we consider a generalization of ERW, called ERW with random step sizes, such that the step sizes are random and varying in time. The ERW with random step sizes was introduced by Fan and Shao [START_REF] Fan | Cramér's moderate deviations for martingales with applications[END_REF]. Let (Z i ) i≥1 be a sequence of positive independent and identically distribution (i.i.d.) random variables, with finite mean ν = 1 and so on, but not on n. This constant may vary from line to line. For two sequences of positive numbers (a n ) n≥1 and (b n ) n≥1 , write a n b n if there exists a constant C > 0 such that a n /C ≤ b n ≤ Ca n for all sufficiently large n. We shall also use the notation a n b n to mean that there exists a positive constant C not depending on n such that a n ≤ Cb n . We also write a n ∼ b n if lim n→∞ a n /b n = 1. We denote by N (0, σ 2 ) the normal distribution with mean 0 and variance σ 2 . N will designate a standard normal random variable, and we will denote by Φ(•) the cumulative distribution function of a standard normal random variable. We shall use the notations ϕ to denote the density of a N (0, 1) distribution and ϕ σ 2 to denote the density of a N (0, σ 2 ) distribution.

Law of iterated logarithm

The almost sure convergence for the ERW has been established by Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] and Bercu and Lucile [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] via a martingale method. In the next theorem, we give the corresponding results for the ERW with random step sizes.

Theorem 2.1. Assume that p ∈ (0, 1] and EZ 2 1 < ∞. [i] If p ∈ (0, 3/4), then lim sup n→∞ |S n | √ n log log n ≤ 2 3 -4p + √ 2σ a.s. (2.1) [ii] If p = 3/4, then lim sup n→∞ |S n | √ n log n log log log n = √ 2 a.s. (2.2) [iii] If p ∈ (3/4, 1], then lim n→∞ S n n 2p-1 = L a.s., (2.3)
where L is a non-degenerate and non Gaussian random variable.

We immediately deduce from Theorem 2.1 that the law of large numbers holds. If p ∈ (0, 1), then lim n→∞ S n n = 0 a.s.

Hence, we find again the Kolmogorov strong law of large numbers given by Coletti, Gava and Schütz [START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF]. See also Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] and Bercu and Lucile [START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF] for closely related results. In particular, Bercu [START_REF] Bercu | On the elephant random walk with stops playing hide and seek with the Mittag-Leffler distribution[END_REF] gave the law of the iterated logarithm for ERW with stops playing hide and seek with the Mittag-Leffler distribution.

Remark 2.1. In [START_REF] Coletti | A strong invariance principle for the elephant random walk[END_REF] a strong invariance principle is proved for the usual ERW, from which the compact law of the iterated logarithm can be easily deduced. We would like to mention that the rate of approximation in this strong invariance principle can be improved by using Theorem 2.1 of Shao [START_REF] Shao | Almost sure invariance principles for mixing sequences of random variables[END_REF]. More precisely, letting v n = n i=1 a 2 i and applying Shao's results, one can prove that [i] If p ≤ 5/8, there exists a standard Wiener process W (t) such that, for any δ > 0,

a n T n -W (v n ) = o a n n 1/4 (log n) 3/4 (log log n) δ+1/4 a.s.
[ii] If 5/8 < p < 3/4, then there exists a standard Wiener process W (t) such that

a n T n -W (v n ) = o u n (log n) 1/2 a.s.
, where u n is any sequence increasing to ∞.

[iii] If p = 3/4, then there exists a standard Wiener process W (t) such that

a n T n -W (v n ) = o u n (log log n) 1/2 a.s.
, where u n is any sequence increasing to ∞.

Normal approximations

In this section, we consider the normal approximation for S n . For p ∈ (0, 3/4] and n ≥ 2, denote

a 1 = 1, a n = Γ(n)Γ(2p) Γ(n + 2p -1) and v n = n i=1 a 2 i .
Here Γ(s) = ∞ 0 t s-1 e -t dt, s > 0, is the Gamma function. Notice that a n and v n are both positive. Moreover, by taking into account the relations (4.2) and (4.3) of Section 4, it is known that v n → ∞ as n → ∞.

Central limit theorem

Theorem 3.1. Assume that EZ 2 1 < ∞. Then, for p ∈ (0, 3/4], Sn

v n + na 2 n σ 2 D -→ N , (3.1)
where Sn = a n S n -(2q -1) and D -→ stands for convergence in distribution.

When p ∈ (0, 3/4] and σ 2 = 0, the central limit theorem for the usual ERW has been established by Coletti, Gava and Schütz [START_REF] Coletti | Central limit theorem and related results for the elephant random walk[END_REF][START_REF] Coletti | A strong invariance principle for the elephant random walk[END_REF] and Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF]. Theorem 3.1 then extends the CLT to the case σ 2 > 0.

Note also that since v n → ∞ as n → ∞, (3.1) also holds if we replace Sn by a n S n (indeed the term (2q -1)/ v n + na 2 n σ 2 goes to 0 as n → ∞). However, it is convenient to consider random variables which are centered as Sn . Indeed if we consider a n S n rather than Sn then the corresponding rates of convergence in the CLT are slower (see for instance Remark 3.1 concerning the Kolmogorov distance).

Concerning the normalizing sequence, note that when p ∈ (0, 3/4), then (4.1) combined with (4.2) entails that

b n := a -1 n v n + na 2 n σ 2 ∼ n(σ 2 + 1/(3 -4p)),
whereas when p = 3/4, (4.1) combined with (4.2) implies that b n ∼ √ n log n. Finally, notice that when p ∈ (3/4, 1], v n + na 2 n σ 2 is bounded and a n is in order of n 1-2p (see (4.1)). Thus, by (2.3), Sn / v n + na 2 n σ 2 converges to a non-degenerate and non Gaussian random random variable, and (3.1) does not hold. Hence, we shall only consider the case p ∈ (0, 3/4] in the remaining of the paper.

Convergence rates in terms of Kologmorov's distance

For a real-valued random variable X, denote

K(X) = sup u∈R P X ≤ u -Φ(u)
the Kologmorov's distance between the law of X and the standard normal distribution.

We start by giving the rates of convergence in the central limit theorem for the Kolmogorov distance and the usual ERW. Theorem 3.2. Let p ∈ (0, 3/4]. Then the following inequality holds:

K a n T n -(2q -1) √ v n 1 √ n + 1 v n .
Taking into account (4.2) and (4.3) of the next section, it follows that when p ≤ 5/8, the rate in Theorem 3.2 is 1/ √ n and when p > 5/8, the rate is 1/v n . Note also that Theorem 3.2 improves on the previous results obtained by Fan et al. [15]

K a n T n √ v n 1 √ n + 1 √ v n .
The following theorem gives a Berry-Esseen type bound for the ERW with random step sizes. Theorem 3.2 can be viewed as a consequence of it (taking ρ = 1 and σ = 0.)

Theorem 3.3. Assume that M (ρ) = EZ 2+ρ 1 < ∞ for a constant ρ ∈ (0, 1].
The following inequality holds:

K a n S n -(2q -1) v n + na 2 n σ 2 1 n ρ/2 + 1 v n .

Convergence rates in terms of Zolotarev's distances

Zolotarev's distance of order r can be described as follows: for two probability measures µ and ν, and any r > 0,

ζ r (µ, ν) = sup f dµ -f dν : f ∈ Λ r ,
where Λ r is defined as follows: denoting by l the natural integer satisfying l < r ≤ l + 1, Λ r is the class of real functions f which are l-times continuous differentiable and such that

|f (l) (x) -f (l) (y)| ≤ |x -y| r-l for any (x, y) ∈ R 2 .
For two random variables X and Y , let µ X and ν Y be respectively the probability laws of X and Y . We denote ζ r (X, Y ) = ζ r (µ X , ν Y ) to soothe the notations.

We start by giving the rates of convergence in the central limit theorem in terms of Zolotarev distances for the usual ERW.

Theorem 3.4. Let r ∈ (0, 2]. Then the following inequality holds:

ζ r a n T n -(2q -1) √ v n , N 1 n r/2 + 1 v n .
Note that ζ 1 = W 1 , where W 1 is the Wasserstein distance of order 1 (see the next section for the definition). Hence, for r = 1, Theorem 3.4 improves on the previous results obtained by Ma et al. [22].

The following theorem gives some bounds for the ERW with random step sizes.

ζ r a n S n -(2q -1) v n + na 2 n σ 2 , N 1 n (r∧ρ)/2 + 1 v n .

Convergence rates in terms of Wasserstein's distances

We first recall the definitions of Wasserstein's distances. Let L(µ, ν) be a set of probability laws on R 2 with marginals µ and ν. The Wassertein distance of order r > 0 is defined as follows:

W r (µ, ν) = inf |x -y| r P(dx, dy)

1/ max(1,r)

: P ∈ L(µ, ν) .

For two random variables X and Y with respective laws µ X and ν Y , denote W r (X, Y ) = W r (µ X , ν Y ) to soothe the notation. For r ∈ (0, 1], using Kantorovich-Rubinstein's theorem with the metric d(x, y) = |x -y| r , we have W r (µ, ν) = ζ r (µ, ν). Hence, for r ∈ (0, 1], the inequality of Theorem 3.5 holds for the distance W r .

Next, for r ∈ [START_REF] Baur | Elephant random walks and their connection to Pólya-type urns[END_REF][START_REF] Bercu | A martingale approach for the elephant random walk[END_REF], from [START_REF] Rio | Upper bounds for minimal distances in the central limit theorem[END_REF], we know that there exists a positive constant c r such that

W r (µ, ν) ≤ c r (ζ r (µ, ν)) 1/r . (3.2)
Therefore, starting from Theorem 3.4, we derive for the usual ERW that for any r

∈ [1, 2], W r a n T n -(2q -1) √ v n , N 1 √ n + 1 v 1/r n .
Combining this last estimate with some precise upper bounds for independent random variables given by Bobkov [START_REF] Bobkov | Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances[END_REF], we get the following result for the ERW with random step sizes.

Theorem 3.6. Let r ∈ [1, 2], and assume that EZ 2+ρ 1 < ∞ for a constant ρ ∈ (0, r]. The following inequalities hold:

W r a n S n -(2q -1) √ v n , N 1 n ρ/2r + 1 v 1/r n .
Note that when ρ ∈ (0, 1], Theorem 3.6 is a direct consequence of Theorem 3.5 combined with inequality (3.2).

Preliminary considerations and lemmas

By the well-known Stirling formula

log Γ(x) = x - 1 2 log x -x + 1 2 log 2π + O 1 x as x → ∞, we deduce that for p ∈ (0, 1], lim n→∞ a n n 2p-1 = Γ(2p). (4.1)
Moreover, for p ∈ (0, 3/4), we have

lim n→∞ v n n 3-4p = Γ(2p) 2 3 -4p , (4.2)
and, for p = 3/4,

lim n→∞ v n log n = π 4 . (4.3)
See also Bercu [2] for the equalities (4.1)-(4.3). Denote

γ n = 1 + 2p -1 n , n ≥ 1.
It is easy to see that a 1 = 1 and for n ≥ 2,

a n = n-1 i=1 1 γ i . Notice that a n+1 = a n /γ n . Thus, a n is increasing in n if 0 < p ≤ 1/2,
a i =      a n if 0 < p ≤ 1/2 (2p) -1 if 1/2 < p ≤ 1. (4.4)
Define the filtration

F 0 = {∅, Ω}, F n = σ{α i , X 1 , β i , Z i : 1 ≤ i ≤ n}, n ≥ 1,

and denote

a n S n -(2q -1) = a n n i=1 α i X βi (Z i -1) + a n T n -(2q -1). (4.5) Let M 0 = 0, M n = a n T n -(2q -1), n ≥ 1.
As noticed in [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF], M n , F n n≥0 is a martingale. Indeed, we have

E[M 1 |F 0 ] = E[a 1 T 1 -(2q -1)] = 0,
and for n ≥ 1,

E[M n+1 |F n ] = E[a n+1 T n+1 -(2q -1)|F n ] = a n+1 T n + a n+1 E[α n+1 ]E[X βn+1 |F n ] -(2q -1) = a n+1 T n + (2p -1) T n n -(2q -1) = a n T n -(2q -1) = M n a.s.
Moreover, we can rewrite (M n , F n ) n≥1 in the following additive form

M n = n i=1 a i ε i := n i=1 ∆M i , (4.6)
where

ε i = T i -γ i-1 T i-1 with the convention γ 0 T 0 = 2q -1. Note that (∆M k , F k ) k≥0 is a sequence of martingale differences such that Var(M n ) = n i=1 a 2 i -(2p -1) 2 Eζ n , where ζ n = n-1 k=1 a 2 k+1 T k k (see (A.7) in [2]). In particular Var(M n ) ≤ v n and a n T n 2 2 ≤ 2(1 + v n ). (4.7)
The following lemma shows that ∆M k is bounded.

Lemma 4.1. For k ≥ 1 and p ∈ (0, 1], it holds |∆M k | ≤ 2a k .
Proof. We have

|∆M 1 | = |a 1 T 1 -(2q -1)| ≤ |X 1 | + |2q -1| ≤ 2 = 2a 1 ,
and, for k ≥ 2,

∆M k = a k T k -a k-1 T k-1 = a k X k -a k T k-1 k -1 (2p -1). Since |X k | ≤ 1, we have |T k | ≤ k. Hence, for k ≥ 2, |∆M k | ≤ a k |X k | + a k |2p -1| ≤ 2a k .
This completes the proof of the lemma.

Next lemma gives estimates of the conditional moments of order 3 and 4 of the sequence of martingale differences (∆M k , F k ) k≥1 . Lemma 4.2. For k ≥ 1, the following equalities hold:

E[∆M 3 k |F k-1 ] = 2(2p -1)a 3 k - T k-1 k -1 + (2p -1) 2 T 3 k-1 (k -1) 3
and

E[∆M 4 k |F k-1 ] = a 4 k + a 4 k (2p -1) 2 2 T 2 k-1 (k -1) 2 -3(2p -1) 2 T 4 k-1 (k -1) 4 ,
with the convention T 0 /0 = 1. In particular,

E[∆M 3 k |F k-1 ] ≤ 4a 3 k T k-1 k -1 and E[∆M 4 k |F k-1 ] -a 4 k ≤ 5a 4 k T 2 k-1 (k -1) 2 .
Proof. The equality for the conditional moment of order 4 comes from [2, (A.5)]. It remains to prove the equality for the conditional moment of order 3. Note first that

E[∆M 3 k |F k-1 ] = a 3 k E[(T k -γ k-1 T k-1 ) 3 |F k-1 ] = a 3 k E[T 3 k |F k-1 ] -3γ k-1 T k-1 E[T 2 k |F k-1 ] + 3γ 2 k-1 T 2 k-1 E[T k |F k-1 ] -γ 3 k-1 T 3 k-1 .
For

E[T 3 k |F k-1 ],
we have the following estimation

E[T 3 k |F k-1 ] = E[(T k-1 + α k X β k ) 3 |F k-1 ] = T 3 k-1 + 3T 2 k-1 E[α k X β k |F k-1 ] +3T k-1 E[(α k X β k ) 2 |F k-1 ] + E[(α k X β k ) 3 |F k-1 ] = T 3 k-1 + 3(2p -1) T 3 k-1 k -1 + 3T k-1 + (2p -1) T k-1 k -1 = (3γ k-1 -2)T 3 k-1 + (γ k-1 + 2)T k-1 . Notice that E[T 2 k |F k-1 ] = (2γ k-1 -1)T 2 k-1 + 1 and E[T k |F k-1 ] = γ k-1 T k-1 (see the relations (A.
3) and (2.3) in [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF]). Hence, we get

E[∆M 3 k |F k-1 ] = 2a 3 k T k-1 (1 -γ k-1 ) + T 3 k-1 (γ k-1 -1) 3 .
The first equality follows from the fact that γ n = 1 + 2p-1 n . To get the inequalities, it suffices to notice that |T k | ≤ k.

Proof of Theorem 2.1

We first give a proof of item [i]. For any n ≥ 1, let

H n = n i=1 α i X βi (Z i -1).
(5.1) We complete the proof by proving item [iii] that is when 3/4 < p ≤ 1. Since in this case n 2p-1 / √ n → 0, (5.2) entails that lim n→∞ Hn n 2p-1 = 0. Hence the result follows from Theorems 3.7 and 3.8 in Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF] and the fact that S n = H n + T n .

By (4.5), recall that S n = H n + T n . Let G n = σ(α i , β i , X 1 , Z i , 1 ≤ i ≤ n). Note that (H n , G n ) n≥1 is a zero-mean, square integrable martingale such that Var(α i X βi (Z i -1)) = σ 2 and |α i X βi (Z i -1)| = |Z i -1|.
6 Proof of Theorem 3.1 By (5.1), we have

a n S n -(2q -1) v n + na 2 n σ 2 = a n H n v n + na 2 n σ 2 + M n v n + na 2 n σ 2 =: U n + V n , (6.1)
where H n is defined in (5.1). Denote F = σ(α i , β i , X 1 , i ≥ 1). For s, t ∈ R, write

f n (s, t) = E exp{(itU n + isV n )} = E E[exp{itU n }|F ] exp{isV n } , the joint characteristic function of (U n , V n ). Since (Z i ) i≥1 is a sequence of i.i.d. r.v.'s, lim n→∞ E[exp{itU n }|F ] = ϕ(σ 1 t) a.s., where σ 2 1 = lim n→∞ na 2 n σ 2 v n + na 2 n σ 2
and ϕ(t) = exp{-t 2 /2}. Now

f n (s, t) = E E[exp{itU n }|F ] -ϕ(σ 1 t) exp{isV n } + ϕ(σ 1 t)E[exp{isV n }].
From the CLT for the usual ERW (see Theorems 3.3 and 3.4 in Bercu [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF]), we know that

lim n→∞ E[exp{isV n }] = ϕ(σ 2 s), where σ 2 2 = lim n→∞ v n v n + na 2 n σ 2 .
Consequently, it holds lim

n→∞ f n (s, t) = ϕ(σ 1 t)ϕ(σ 2 s).
In particular, since σ 2 1 + σ 2 2 = 1, we get lim n→∞ f n (t, t) = ϕ(t), which implies that (U n + V n ) n≥1 converges in distribution to a standard normal random variable.

Proof of Theorem 3.2

The proof of Theorem 3.2 is a refinement on the argument of Lemma 3.3 of Grama and Haeusler [START_REF] Grama | Large deviations for martingales via Cramér's method[END_REF], where the authors obtained the best possible Berry-Esseen bound for martingales with bounded differences. Compared to [START_REF] Grama | Large deviations for martingales via Cramér's method[END_REF], the main challenge of our proof comes from the fact that the conditional variance of martingales does not converge to a constant in L ∞ -norm. Our proof will be based on Lindeberg's telescoping sums argument, see Bolthausen [START_REF] Bolthausen | Exact convergence rates in some martingale central limit theorems[END_REF]. The following two technical lemmas due to Bolthausen [START_REF] Bolthausen | Exact convergence rates in some martingale central limit theorems[END_REF] will be needed.

sup u∈R P (X ≤ u) -Φ (u) ≤ c 1 sup u∈R P (X + Y ≤ u) -Φ (u) + c 2 E Y 2 |X 1/2 ∞ ,
where c 1 = 2 and c 2 = 5/ √ 2π.

Lemma 7.2. [8, Lemma 2] Let G be an integrable function on R of bounded variation G V , X be a random variable and a, b = 0 be real numbers. Then

E G X + a b ≤ G V sup u P (X ≤ u) -Φ (u) + G 1 |b|, where G 1 is the L 1 (R) norm of G.
We shall divide the proof in two steps according to the values of p.

1) Case p ∈ (0, 3/4). Let us introduce some notations that will be used all along the proof. Let β be a constant greater than one that will be specified later. Denote

κ 2 n = max [n/2]≤i≤n a 2 i , δ 2 n = βκ 2 n /v n , (7.1)
and for 0

≤ k ≤ n, t k,n = βκ 2 n + v n -v k 1/2 , A k = t 2 k,n /v n , u k,n = β + β -1 (1 + β) n i=k+1 a 2 i /κ 2 n . (7.2)
Note that A k is non-increasing in k, and satisfies A 0 = δ 2 n + 1 and A n = δ 2 n . Moreover, for u, x ∈ R and y > 0, set, for brevity,

Φ u (x, y) = Φ u -x √ y . (7.3)
Let N 1 be a standard normal random variable, which is also independent of

T n . For 1 ≤ k ≤ n, denote K0 = 0, Kk = 1 √ v n a k T k -(2q -1) , ξ k = Kk -Kk-1 and σ 2 k = a 2 k v n .
Then we have

a n T n -(2q -1) √ v n = Kn = n k=1 ξ k .
Next, we estimate the upper bound of K( Kn ). By Lemma 7.1, we get

K( Kn ) ≤ c 1 sup u∈R P Kn +δ n N 1 ≤ u -Φ(u) +c 2 δ n = c 1 sup u∈R E[Φ u ( Kn , A n )]-Φ(u) +c 2 δ n . Hence K( Kn ) ≤ c 1 sup u∈R E[Φ u ( Kn , A n )]-E[Φ u ( K0 , A 0 )] + c 1 sup u∈R Φ u δ 2 n + 1 -Φ(u) +c 2 δ n .
There exist positive constants c 3 and c 4 such that for any n ≥ 1,

Φ u δ 2 n + 1 -Φ(u) ≤ c 3 1 δ 2 n + 1 -1 ≤ c 4 δ 2 n . (7.4) Therefore, since δ 2 n ≤ β, K( Kn ) ≤ c 1 sup u∈R E[Φ u ( Kn , A n )] -E[Φ u ( K0 , A 0 )] + c 5 δ n . (7.5)
Next, we give an estimation for

E[Φ u ( Kn , A n )] -E[Φ u ( K0 , A 0 )]. Write first E[Φ u ( Kn , A n )] -E[Φ u ( K0 , A 0 )] = n k=1 E Φ u ( Kk , A k ) -Φ u ( Kk-1 , A k-1 ) .
Using the fact that

∂ 2 ∂x 2 Φ u (x, y) = 2 ∂ ∂y Φ u (x, y),
and that E[ξ k |F k-1 ] = 0, we obtain

E[Φ u ( Kn , A n )] -E[Φ u ( K0 , A 0 )] = I 1 + I 2 -I 3 ,
where

I 1 = n k=1 E Φ u ( Kk , A k ) -Φ u ( Kk-1 , A k ) - ∂ ∂x Φ u ( Kk-1 , A k )ξ k - 1 2 ∂ 2 ∂x 2 Φ u ( Kk-1 , A k )ξ 2 k , I 2 = 1 2 n k=1 E ∂ 2 ∂x 2 Φ u ( Kk-1 , A k ) E[ξ 2 k |F k-1 ] -σ 2 k , I 3 = n k=1 E Φ u ( Kk-1 , A k-1 ) -Φ u ( Kk-1 , A k ) - ∂ ∂y Φ u ( Kk-1 , A k )σ 2 k .
From (7.5), we deduce that

K( Kn ) ≤ C |I 1 | + |I 2 | + |I 3 | + δ n . (7.6)
In the sequel, we give some estimates for I 1 , I 2 and I 3 . The notation ϑ i stands for some values or random variables satisfying 0 ≤ ϑ i ≤ 1.

a) Control of I 1 . To shorten notations, denote Ĥk-

1 (u) = u -Kk-1 √ A k . For 1 ≤ k ≤ n,
we have

R k := Φ u ( Kk , A k ) -Φ u ( Kk-1 , A k ) - ∂ ∂x Φ u ( Kk-1 , A k )ξ k - 1 2 ∂ 2 ∂x 2 Φ u ( Kk-1 , A k )ξ 2 k = Φ Ĥk-1 (u) - ξ k √ A k -Φ( Ĥk-1 (u)) + Φ ( Ĥk-1 (u)) ξ k √ A k - 1 2 Φ ( Ĥk-1 (u)) ξ k √ A k 2 .
By Taylor expansion at order 4, we deduce that

I 1 = n k=1 ER k,1 + n k=1 ER k,2 , (7.7)
where

R k,1 = - 1 6 Φ Ĥk-1 (u) ξ k √ A k 3 and R k,2 = 1 24 Φ (4) Ĥk-1 (u) -ϑ k ξ k √ A k ξ k √ A k 4 .
Next, we handle the term

|ER k,1 |. As Ĥk-1 (u) is F k-1 -measurable, we have ER k,1 ≤ 1 A 3/2 k E Φ Ĥk-1 (u) E[ξ 3 k |F k-1 ] ≤ C 1 A 3/2 k E E[ξ 3 k |F k-1 ] .
By Lemma 4.2, we have Eξ 3

1 ≤ 4a 3 1 /v 3/2
n , and for 2 ≤ k ≤ n,

E E ξ 3 k F k-1 a k √ v n 3 E T k-1 k -1 a 3 k (E a k-1 T k-1 2 ) 1/2 a k-1 v 3/2 n (k -1)
.

Taking into account (4.7), it follows that for 2 ≤ k ≤ n,

E E ξ 3 k F k-1 a 3 k (1 + √ v k-1 ) a k-1 v 3/2 n (k -1)
.

Hence, we get

n k=1 ER k,1 1
v 3/2 n + n k=2 1 A 3/2 k a 3 k √ v k-1 a k-1 v 3/2 n (k -1) . ( 7.8)
Next we give an estimation for the last bound. By (4.1) and (4.2), and since p ∈ (0, 3/4),

√ v k-1 a k-1 (k -1) = O √ k , k → ∞. Note also that A k v n = t 2 k,n and, since p ∈ (0, 3/4), max 1≤k≤[n/2] t -2 k,n = t -2 [n/2],n v -1 n . Therefore, we get n k=2 1 A 3/2 k a 3 k √ v k-1 a k-1 v 3/2 n (k -1) v -3/2 n k∈[1,n/2] a 3 k √ k + κ n √ n k∈(n/2,n] t -3 k,n a 2 k . (7.9)
For the first term on the right-hand side of (7.9), we infer that

v -3/2 n k∈[1,n/2] a 3 k √ k v -3/2 n 1 p>7/12 + v -3/2 n (log n)1 p=7/12 + n -1 1 p<7/12 . (7.10)
To deal with the second term on the right-hand side of (7.9), we first notice that for any integer

k such that [n/2] ≤ k ≤ n, a 2 k t 2 k,n ≤ log u k-1,n /u k,n and κ 2 n u k-1,n 3 ≤ t 2 k,n ≤ κ 2 n u k,n (7.11)
(see inequalities (5.2) and (5.4) in [START_REF] Dedecker | Rates of convergence in the central limit theorem for martingales in the non stationary setting[END_REF]). Therefore, for any α ≥ 0,

T n (α) := n k=[n/2]+1 t α k,n a 2 k t 2 k,n ≤ κ α n n k=[n/2]+1 u α/2 k,n u k-1,n u k,n 1 x dx ≤ κ α n n k=[n/2]+1 u k-1,n u k,n
x α/2-1 dx and, for any α < 0,

T n (α) ≤ κ α n 3 -α/2 n k=[n/2]+1 u α/2 k-1,n u k-1,n u k,n 1 x dx ≤ κ α n 3 -α/2 n k=[n/2]+1 u k-1,n u k,n
x α/2-1 dx.

14 So, overall, for any real α, using the fact that p ∈ (0, 3/4), we get

T n (α) = n k=[n/2]+1 t α k,n a 2 k t 2 k,n ≤ κ α n (1 + 3 -α/2 ) u [n/2],n β x α/2-1 dx κ α n β α/2 1 α<0 + (log n)1 α=0 + v α/2 n 1 α>0 . (7.12)
Therefore, by (7.12) applied with α = -1, we get

κ n √ n k∈(n/2,n] t -3 k,n a 2 k n -1/2 . ( 7
ξ k √ A k ≤ min 2/ βκ 2 n , 2/ v n -v [n/2]
. So, in each case, we can select β large enough in such a way that

ξ k √ A k ≤ 1.
From now on, β is selected this way. Thus we have

|R k,2 | ≤ 1 24 Φ (4) Ĥk-1 (u) -ϑ k ξ k √ A k ξ k √ A k 4 ≤ G( Ĥk-1 (u)) ξ k √ A k 4 ,
where

G(z) = sup |t-z|≤1 |Φ (4) t |. By Lemma 4.1, ξ 4 k ≤ 2 4 a 4 k v -2 n . Therefore, n k=1 |ER k,2 | n k=1 E[G( Ĥk-1 (u))] A 2 k a k √ v n 4 .
By the definition of Ĥk-1 (u) and Lemma 7.2, it follows that there exists a positive constant C such that for any k ≥ 1 and any u ∈ R,

E[G( Ĥk-1 (u))] ≤ C K Kk-1 + C A k . Now, by Lemma 7.1, K Kk-1 ≤ CK Kn + C E[( Kn -Kk-1 ) 2 | Kk-1 ] 1/2 ∞ ,
and, by the martingale property of (M n , F n ) n≥1 and Lemma 4.1,

E[( Kn -Kk-1 ) 2 | Kk-1 ] = v -1 n n i=k E[(∆M i ) 2 | Kk-1 ] ≤ 4v -1 n (v n -v k-1 ).
Therefore, it holds

E[G( Ĥk-1 (u))] K Kn + A k-1 . (7.15)
So, overall, we have

n k=1 ER k,2 n k=1 1 A 2 k a k √ v n 4 K Kn + n k=1 1 A 3/2 k-1 a k √ v n 4 . (7.16)
Next, we give some estimations for the right-hand side of the last inequality. Recall that max 1≤k≤

[n/2] (A k v n ) -1 = t -2 [n/2],n v -1 n . Therefore n k=1 1 A 2 k a k √ v n 4 v 2 n [n/2] k=1 a 4 k + κ 2 n n k=[n/2]+1 t -2 k,n a 2 k t 2 k,n . Now n k=1 a 4 k 1 p>5/8 + (log n)1 p=5/8 + a 2 n v n 1 p<5/8
. On another hand, by taking into account (7.12) with α = -2, we get

n k=[n/2]+1 t -2 k,n a 2 k t 2 k,n κ -2 n β -1 .

It follows that

n k=1 1 A 2 k a k √ v n 4 β -1 + n -1 + v -2 n log n.
Using similar arguments, we infer that

n k=1 1 A 3/2 k a k √ v n 4 v -2 n n k=1 a 4 k + v -1/2 n κ n v -2 n 1 p>5/8 + (log n)1 p=5/8 + n -1/2 .
It follows that, for n large enough,

n k=1 ER k,2 β -1 K Kn + v -2 n 1 p>5/8 + n -1/2 . (7.17)
Starting from (7.7) and taking into account (7.14) and (7.17), we derive that

|I 1 | β -1 K Kn + 1 v 3/2 n + 1 √ n . (7.18) b) Control of I 2 .
We have

|I 2 | ≤ n k=1 1 2A k E ϕ ( Ĥk-1 (u)) E[ξ 2 k |F k-1 ] -σ 2 k n k=1 1 A k E E[ξ 2 k |F k-1 ] -σ 2 k .
Clearly, we have

E[ξ 2 1 |F 0 ] = σ 2 1 .
On another hand, by relation (A.4) in [START_REF] Bercu | A martingale approach for the elephant random walk[END_REF], we get, for 2

≤ k ≤ n, E[∆M 2 k |F k-1 ] = a 2 k -a 2 k (2p -1) 2 T k-1 k -1 so that E[ξ 2 k |F k-1 ] -σ 2 k ≤ (2p -1) 2 a 2 k v n T k-1 k -1 2 .
Thus, by (4.7),

|I 2 | n k=2 1 A k a 2 k v n a 2 k-1 (k -1) 2 E(a k-1 T k-1 ) 2 k∈[2,n/2] 1 A k a 2 k v n v k-1 a 2 k-1 (k -1) 2 + k∈(n/2,n] 1 A k a 2 k v n v k-1 a 2 k-1 (k -1) 2 .
By and (4.2), we have for p ∈ (0, 3/4),

v k-1 a 2 k-1 (k -1) 2 1 k , k → ∞, and max 1≤k≤[n/2] (A k v n ) -1 v -1 n .
Hence

k∈[2,n/2] 1 A k a 2 k v n v k-1 a 2 k-1 (k -1) 2 v -1 n k∈[2,n/2] a 2 k k n -1 + v -1 n + n -1 (log n)1 p=1/2 .
On another hand, by (7.12) with α = 0, we get

k∈(n/2,n] 1 A k a 2 k v n v k-1 a 2 k-1 (k -1) 2 n -1 k∈(n/2,n] a 2 k t 2 k,n n -1 log n.
So, overall,

|I 2 | log n n + 1 v n . (7.20) c) Control of I 3 . Note first that Φ u ( Kk-1 , A k-1 ) -Φ u ( Kk-1 , A k ) - ∂ ∂y Φ u ( Kk-1 , A k )σ 2 k = 1 2 ∂ 2 ∂y 2 Φ u ( Kk-1 , A k + ϑ k σ 2 k )σ 4 k , for some ϑ k ∈ [0, 1]. Now, ∂ 2 ∂y 2 Φ u (x, z) = 3 4 u -x z 5/2 ϕ u -x √ z + 1 4 (u -x) 2 z 3 ϕ u -x √ z and, for any ϑ ∈ [0, 1], A k-1 /5 ≤ A k +ϑσ 2 k ≤ A k-1 .
The right-hand side of this inequality is trivial. To prove the left hand side, we note that a k ≤ 2a k+1 which entails that a

2 k ≤ 4 n =k+1 a 2 . Therefore, v n A k-1 ≤ βκ 2 n +5 n =k+1 a 2 ≤ 5v n A k . Let g(t) = (|t|+|t| 3 )ϕ(t). For t = (u -Kk-1 )/ A k + ϑσ 2 k and H k-1 (u) = (u -Kk-1 )/ A k-1 , we then get g(t) ≤ √ 5| H k-1 (u)| + 5 3/2 | H k-1 (u)| 3 ϕ( H k-1 (u)) := G( H k-1 (u)).
So, overall,

|I 3 | ≤ n k=1 1 A 2 k E[ G( H k-1 (u))] σ 4 k .
Proceeding as to get (7.15), we infer that

E[ G( H k-1 (u))] K Kn + A k-1 . Therefore |I 3 | n k=1 1 A 2 k σ 4 k K Kn + n k=1 1 A 3/2 k σ 4 k .
Hence the right-hand side of the above inequality is the same as the one of inequality (7.16). Therefore, according to (7.17), we get, for n large enough, that

|I 3 | β -1 K Kn + v -2 n 1 p>5/8 + n -1/2 . (7.21)
Starting from (7.6) and taking into account the estimates (7.18), (7.20), (7.21) and the fact that δ n v -1 n +n -1 , it follows that there exists a positive constant C (not depending on n) such that, for n large enough,

(1 -Cβ -1 )K Kn n -1/2 + v -1 n ,
and Theorem 3.2 (for p ∈ (0, 3/4)) follows by taking β large enough so that 1 -

Cβ -1 ≥ 1/2.
2) Case p = 3/4. Recall that in this case, v n /(log n) → π/4. Compared to the previous case, the differences are as follows. First we fix a γ ∈ (0, 1) and we define

κ 2 n = max [n γ ]≤i≤n a 2 i = a 2 [n γ ] . (7.22)
Next, δ 2 n , t k,n , A k and u k,n are still defined as in (7.1) and (7.2). It follows that max

1≤k≤[n γ ] (A k v n ) -1 = max 1≤k≤[n γ ] t -2 k,n = t -2 [n γ ],n v -1 n .
On another hand, for any real α,

T n (α) = n k=[n γ ]+1 t α k,n a 2 k t 2 k,n ≤ κ α n (1 + 3 -α/2 ) t [n γ ],n β x α/2-1 dx κ α n β α/2 1 α<0 + (log log n)1 α=0 + v α/2 n 1 α>0 . (7.23)
Moreover we shall use in case p = 3/4 that

√ v k-1 a k-1 (k -1) = O √ log k √ k , k → ∞.
Taking into account all the differences pointed above, separating the sums from 1 to n into a sum from 1 to [n γ ] plus a sum from [n γ ] + 1 to n, and proceeding as for the case p ∈ (0, 3/4) we infer that

|I 1 | + |I 2 | + |I 3 | β -1 K Kn + v -1
n , which completes the proof of Theorem 3.2 in case p = 3/4.

Proof of Theorem 3.3

Now we are in the position to prove Theorem 3.3. Set any integer k in

[1, n], set τ 2 k = na 2 n σ 2 + v k . For 1 ≤ k ≤ n, denote H k = 1 τ n k i=1 α i X βi (Z i -1) and K k = 1 τ n a k T k -(2q -1) .
Then we have

a n S n -(2q -1) v n + na 2 n σ 2 = a n H n + K n .
By the triangle inequality, we get

K a n S n -(2q -1) v n + na 2 n σ 2 ≤ P 1 + P 2 , (8.1)
where

P 1 = sup u∈R P a n H n + K n ≤ u -P N 1 √ na n σ τ n + K n ≤ u and P 2 = sup u∈R P N 1 √ na n σ τ n + K n ≤ u -Φ(u) ,
where N 1 is a standard normal random variable independent of (α k , β k , X k , Z k , k ≥ 0). We first give an upper bound of

P 1 . Let F = σ{α k , β k , X 1 , k ≥ 0}.
By the classical Berry-Esseen bound for independent random variables (see [START_REF] Esseen | On the Liapounoff limit of error in the theory of probability[END_REF]), we have for ρ ∈ (0, 1],

P a n H n + K n ≤ u F -P N 1 √ na n σ τ n + K n ≤ u F = P H n τ n σ √ n ≤ (u -K n )τ n σ √ na n F -P N 1 ≤ (u -K n )τ n σ √ na n F ≤ C n ρ/2 ,
where C is a positive constant depending on σ 2 and EZ 2+ρ 1

. Hence, for ρ ∈ (0, 1],

P 1 ≤ C n ρ/2 . (8.2)
On another hand, since N 1 is independent of K n , we have

P 2 = sup u∈R P K n ≤ u -Φ uτ n √ v n = K K n τ n √ v n 1 √ n + 1 v n , (8.3)
where the last inequality comes from Theorem 3.2. Starting from (8.1) and considering the upper bounds (8.2) and (8.3), we obtain the desired estimate.

9 Proof of Theorem 3.4

Let (N i ) 1≤i≤n be a sequence of N (0, a 2 i )-distributed independent random variables. Assume moreover that (N i ) 1≤i≤n is independent of (∆M i ) 1≤i≤n (recall (4.6) for the defini-

tion of ∆M i ). For n ≥ 3, set U k = k j=1 N j , 1 ≤ k ≤ n.
As in the proof of Theorem 3.2, we shall divide the proof in two cases according to the values of p. 1) Case p ∈ (0, 3/4). We shall use the same notations as those defined in (7.1) and (7.2), selecting β = 1. So

κ 2 n = max [n/2]≤i≤n a 2 i , t k,n = κ 2 n + v n -v k 1/2 , u k,n = 1 + 2 n i=k+1 a 2 i /κ 2 n , 0 ≤ k ≤ n. (9.1)
For any r > 0 and any p ∈ [0, 1], let also

B n (r, p) = 1 n r/2 + 1 v n .
Let N be a N (0, κ 2 n )-distributed random variable independent of (∆M i ) 1≤i≤n and (N i ) 1≤i≤n . Using Lemma 5.1 in Dedecker et al. [START_REF] Dedecker | Rates of convergence for minimal distances in the central limit theorem under projective criteria[END_REF], for any r ∈ (0, 2], we first write

f ∈Λr E[f (M n + N ) -f (U n + N )].
With this aim, we shall use the Lindeberg method and denote

f n,k (x) = E f x + U n -U k + N , 1 ≤ k ≤ n.
By independence of the sequences,

E f (M n + N ) -f n j=1 N j + N = n k=1 D k , (9.3)
where

D k = E f n,k (M k-1 + ∆M k ) -f n,k (M k-1 + N k ) .
Using twice a Taylor's expansion at order 5, we get

D k = I k,1 + I k,2 + I k,3 + I k,4 + I k,5 , (9.4)
where, for any integer i ∈ [START_REF] Baur | Elephant random walks and their connection to Pólya-type urns[END_REF][START_REF] Bercu | On the multi-dimensional elephant random walk[END_REF],

I k,i = 1 i! E f (i) n,k (M k-1 )∆M i k -f (i) n,k (M k-1 )N i k ,
and

I k,5 = 1 120 E f (5) n,k (M k-1 + ϑ 1 ∆M k )∆M 5 k -f (5) n,k (M k-1 + ϑ 2 N k )N 5 k ,
with ϑ 1 , ϑ 2 some random variables with values in [0, 1]. In the sequel, we give some estimations for

n k=1 I k,l , l = 1, 2, • • • , 5. Since (M k , F k ) is a martingale, I k,1 = E f n,k (M k-1 )E[∆M k -N k |F k-1 ] = 0. There- fore, n k=1 I k,1 = 0. (9.5)
We handle now the sum of the |I k,2 |'s. By Lemma 6.1 in [START_REF] Dedecker | Rates of convergence for minimal distances in the central limit theorem under projective criteria[END_REF], for any positive integer i such that i ≥ r, there exists a positive positive constant C r,i such that for any integer n ≥ 1 and any k ∈ [1, n],

f (i) n,k (•) ∞ ≤ C r,i t r-i k,n . (9.6)
For k = 1, we have

I 1,2 = E 1 2 f n,1 (M 0 ) E[∆M 2 1 |F 0 ] -a 2 1 = 0.
On another hand, by taking into account (7.19), (9.6) and (4.7), we get for 2 ≤ k ≤ n,

|I k,2 | = E 1 2 f n,k (M k-1 ) E[∆M 2 k |F k-1 ] -a 2 k (2p -1) 2 t r-2 k,n a 2 k a 2 k-1 E a k-1 T k-1 k -1 2 (2p -1) 2 t r-2 k,n a 2 k a 2 k-1 (1 + v k-1 ) (k -1) 2 . ( 9
v k-1 a 2 k-1 (k -1) 2 1 k , k → ∞. Therefore n k=1 |I k,2 | max 1≤k≤n/2 t r-2 k,n k∈[1,n/2] a 2 k k + 1 n k∈(n/2,n] t r k,n a 2 k t 2 k,n
.

Clearly max 1≤k≤n/2 t r-2 k,n ≤ Cv (r-2)/2 n
. Moreover, by (4.1),

k∈[1,n] k -1 a 2 k 1 + (log n)1 p=1/2 + a 2 n 1 p<1/2 .
Hence, by taking in addition into account (4.4), we derive that, when p = 1/2,

n k=1 |I k,2 | v (r-2)/2 n (1 + a 2 n 1 p<1/2 ) + 1 n k∈(n/2,n] t r k,n a 2 k t 2 k,n
.

Hence, taking into account (7.12) with α = r, we derive that, for any p in (0, 1/2) ∪ (1/2, 3/4),

n k=1 |I k,2 | v r/2 n (v -1 n + n -1 ) + n -1 κ r n u r/2 [n/2],n . But κ r n u r/2 [n/2],n v r/2
n . So, overall, when p ∈ (0, 3/4) and r ∈ (0, 2],

n k=1 |I k,2 | v r/2 n B n (r, p). (9.8)
We handle now the sum of the |I k,3 |'s. By (9.6), Lemma 4.2 and (4.7), we have

|I k,3 | ≤ 1 6 E f n,k (M k-1 )E[∆M 3 k |F k-1 ] -f n,k (M k-1 )E[N 3 k |F k-1 ] t r-3 k,n E E[∆M 3 k |F k-1 ] t r-3 k,n a 3 k E T k-1 k -1 t r-3 k,n a 3 k (k -1)a k-1 E(a k-1 T k-1 ) 2 t r-3 k,n a 3 k √ v k-1 (k -1)a k-
|I k,4 | = 1 24 E f (4) n,k (M k-1 )E[∆M 4 k |F k-1 ] -f (4) n,k (M k-1 )E[N 4 k ] t r-4 k,n a 4 k .
Thus, by taking into account (7.12), we have for p ∈ (0, 3/4] and r ∈ (0, 2],

n k=1 |I k,4 | v (r-4)/2 n [n/2] k=1 a 4 k + κ 2 n n k=[n/2]+1 t r-4 k,n a 2 k v (r-4)/2 n [n/2] k=1 a 4 k + κ r n 1 r∈(0,2) + (log n)1 r=2 .
Thus, for p ∈ (0, 3/4) and r ∈ (0, 2], we infer that

n k=1 |I k,4 | v r/2 n B n (r, p)1 r =2 + κ 2 n (log n)1 r=2 . (9.14)
Therefore, starting from (9.3) and taking into account (9.4), (9.5) and the upper bounds (9.8), (9.12), (9.13) and (9.14), the result is proved for r ∈ (0, 2). It remains to prove it when r = 2. With this aim, it is enough to prove that when r = 2, (2, p). This means to delete the additional logarithmic term log n in the righthand side of (9.14). From now on, we assume that r = 2. Note first that the previous computations proved that

n k=1 |I k,4 | ≤ Cv n B n
] = 3a 4 k , |I k,4 | = 1 24 E f (4) n,k (M k-1 ) E[∆M 4 k |F k-1 ] -E[N 4 k ] ≤ I (1)
k,4 + I

k,4 , where I

k,4 :=

1 24 E f (4) n,k (M k-1 ) E[∆M 4 k |F k-1 ] -a 4 k and I (2) k,4 := a 4 k 12 E f (4)
n,k (M k-1 ) .

Since r = 2, by using (9.6) and Lemma 4.2, we get that

n k=[n/2]+1 I (1) k,4 n k=[n/2]+1 t -2 k,n a 4 k E(T 2 k-1 ) (k -1) 2 n k=[n/2]+1 t -2 k,n a 4 k a 2 k-1 v k-1 (k -1) 2 . As k → ∞, recall that, when p = 3/4, v k-1 a 2 k-1 (k -1) 2 1 k . Therefore n k=[n/2]+1 I (1) k,4 n k=[n/2]+1 t -2 k,n a 4 k k n -1 κ 2 n k∈[n/2,n] a 2 k t 2 k,n .
Hence, taking into account (7.12), we derive that

n k=[n/2]+1 I (1) k,4 n -1 κ 2 n (log n) κ 2 n .
Note that when p ∈ (0, 3/4),

κ 2 n v n /n v n B n (2, p).
We then infer that for r = 2 and any p ∈ (0, 3/4),

n k=[n/2]+1 I (1) k,4 v n B n (2, p) .
Therefore to end the proof of (9.15) and then of the theorem, we need to prove that

n k=[n/2]+1 I (2) k,4 v n B n (2, p) . (9.16)
To achieve this, we need to have a better control of the term E f

n,k (M k-1 ) and we write E f

,k (M k-1 ) = E f (4) n,k (M k-1 ) -E f (4) n,k (G k-1 ) + E f (4) n,k (G k-1 ) , where G k-1 is a normal random variable with variance a 2 1 + • • • + a 2 k-1 . By definition of f n,k , we have E f (4) n,k (N k-1 ) = g (4) n
n,k (0) , where

g n,k (x) = E f x + n i=1,i =k N i + Z . Since, for any integer k such that [n/2] + 1 ≤ k ≤ n, κ 2 n + n i=1,i =k a 2 i ≥ v n , it follows from (9.6) that |g (4) n,k (0)| v -1 n . Therefore, E f (4) n,k (G k-1 ) v -1 n .
Hence, since p ∈ (0, 3/4),

n k=[n/2]+1 a 4 k E f (4) n,k (G k-1 ) 1 v n n k=[n/2]+1 a 4 k a 2 n n -1 v n . (9.17)
We now deal with the term

E f (4) n,k (M k-1 ) -E f (4) n,k (G k-1 ) = E f (M k-1 -u) -E f (G k-1 -u) ϕ (3) t 2 n,k (u)du.
Hence, we have

E f (4) n,k (M k-1 ) -E f (4) n,k (G k-1 ) ≤ c sup u∈R E f (M k-1 -u) -E f (G k-1 -u) ϕ (3) t 2 n,k (u) du.
Recall that we have proved Theorem 3.4 for any r ∈ (0, 2). Hence applying it for r = 1, we have, for any integer

k such that [n/2] + 1 ≤ k ≤ n, sup u∈R E f (M k-1 -u) -E f (G k-1 -u) v n n + 1 √ v n .
On another hand,

ϕ (3) t 2 n,k (u) du t -3 k,n .
Consequently, by taking into account (7.12),

n k=[n/2]+1 a 4 k E f (4) n,k (M k-1 ) -E f (4) n,k (G k-1 ) κ 2 n v n n + 1 √ v n n k=[n/2]+1 a 2 k t 3 k,n κ n v n n + 1 √ v n v n 1 n + 1 v n √ n .
Hence, for any p ∈ (0, 3/4),

n k=[n/2]+1 a 4 k E f (4) n,k (M k-1 ) -E f (4) n,k (G k-1 ) v n B n (2, p). (9.18)
Combining the inequalities (9.17) and (9.18) together (9.16) follows. This completes the proof of the theorem when p ∈ (0, 3/4).

2) Case p = 3/4. As in the proof of Theorem 3.2 in case p = 3/4, we fix γ ∈ (0, 1) and we select κ 2 n as in (7.22). The definitions of t k,n and u k,n are as in (9.1). Recall also that v n / log n → π/4. We start again from the inequality (9.2) and use the decompositions (9.3) and (9.4). We still have to get upper bounds for the quantities .

|I k,i | for i = 2, • • • , 5. For |I k,2 | compared to the previous case, the difference is that v k-1 a 2 k-1 (k -1) 2 log k k , k → ∞.
We then infer that when r ∈ (0, 2] and p = 3/4, Considering all these upper bounds, Theorem 3.2 is then proved in case p = 3/4.

10 Proof of Theorem 3.5

We start from the decomposition (6.1) of Section 6. Let also Let then G 1,n and G 2,n be two independent normal random variables with respective variances σ 2 1,n and σ 2 2,n , and independent of (α i , β i , X 1 , Z i ) i≥1 . By the triangle inequality,

ζ r (U n + V n , N ) ≤ ζ r (U n + V n , G 1,n + V n ) + ζ r (G 1,n + V n , G 1,n + G 2,n ) . (10.2)
We first deal with the first term on right hand in (10.2). Let F = σ(α i , β i , X 1 , i ≥ 1), and note that Applying Theorem 2.1 in [START_REF] Dedecker | Rates of convergence in the central limit theorem for martingales in the non stationary setting[END_REF] if (r, ρ) = (1, 1) and the result of Bikelis [START_REF] Bikelis | Estimates of the remainder term in the central limit theorem[END_REF] if (r, ρ) = (1, 1), we infer that, for any r ∈ (0, 2] and ρ ∈ (0, 1],

ζ r (U n + V n , G 1,n + V n ) ≤ E[ζ
ζ r (P Un|F , P G1,n ) ≤ C na 2 n σ 2 v n + na 2 n σ 2 r 1 n (r∧ρ)/2 a.s. (10.5) where C is a constant depending only on r, σ 2 and EZ 2+ρ We now deal with the second term on the right hand side of (10.2). Since G 1,n is independent of (V n , G 2,n ), it is easy to see that

ζ r (G 1,n + V n , G 1,n + G 2,n ) ≤ ζ r (V n , G 2,n ) . Notice that ζ r (V n , G 2,n ) = √ v n v n + na 2 n σ 2 r ζ r a n T n -(2q -1) √ v n , N ≤ ζ r a n T n -(2q -1) √ v n , N .
Thus from Theorem 3.4, we get for r ∈ (0, 2],

ζ r (G 1,n + V n , G 1,n + G 2,n ) 1 n r/2 + 1 v n . ( 10.7)
Starting from (10.2) and considering the upper bounds (10.6) and (10.7), the result follows.

Theorem 3 . 5 . 1 <

 351 Assume that EZ 2+ρ ∞ for a constant ρ ∈ (0, 1]. Let r ∈ (0, 2]. The following inequality holds:

Lemma 7 . 1 .

 71 [8, Lemma 1] Let X and Y be random variables. Then

2 n /n r/ 2 .

 22 ζ r (P Mn , P Un ) ≤ 2ζ r (P Mn+N , P Un+N ) It remains to give an estimation for ζ r (P Mn+N , P Un+N ) = sup

. 7 )

 7 Clearly, when p = 1/2, |I k,2 | = 0 for any positive integer k. In what follows we handle n k=1 |I k,2 | when p ∈ (0, 3/4)\{1/2}. In this case, by (4.1) and (4.2), we have

. 1 ( 2 n 1 ( 1 , 2 ]..

 12112 We handle now the term n k=1 |I k,3 |. When, p = 3/4, the only difference with the previous case is that√ v k-B n = k∈(n γ ,n] t r-3 k,n a 3 k . Since p = 3/4, A n v (r-3)/2 n. On another hand, when p = 3/4, by using (7.23), we infer thatB n κ r n 1 (0,1) (r) + κ n (log log n)1 r=1 + κ n v (r-1)/Therefore, when p = 3/4, for any r in (0, 2],n k=1 |I k,3 | v (r-2)/2 nWe handle now the term n k=1 |I k,4 |. We write this time By using(7.23), it follows that, for any r ∈ (0, 2], taking into account (7.23), we have for 0 < r ≤ 2,

n σ 2 v n + na 2 n σ 2 and σ 2 2,n = v n v n + na 2 n σ 2 .

 22

1 .

 1 Hence, by (10.3),(10.4) and (10.5), we getζ r (U n + V n , G 1,n + V n ) 1 n (r∧ρ)/2 .(10.6)

  Using Lemma 7.1, we then derive that for the non centered quantity a n T n / √ v n we have the following rate of convergence in the CLT

	Remark 3.1.
	the rate (log n)/ √	n when p ≤ 1/2, (log n)/	who obtained √ v n when p ∈ (1/2, 3/4) and (log log n)/ √ v n
	when p = 3/4.

  Hence starting from (9.7), we get in this case that

	n k=1	|I k,2 |	max 1≤k≤n γ t r-2 k,n	k∈[1,n γ ]	log k k 2 +	log n n γ	k∈(n γ ,n]	t r k,n	a 2 k t 2 k,n

  r (P Un+Vn|F , P G1,n+Vn|F )] .(10.3) Now, since V n is F-measurable and G 1,n is independent of F, we have ζ r (P Un+Vn|F , P G1,n+Vn|F ) = ζ r (P Un|F , P G1,n ) a.s.(10.4)Conditionally to F, U n is a sum of independent random variables with variance a 2 n σ 2 /(v n + na 2 n σ 2 ) and absolute moment of order 2 + ρ bounded byE[ |Z 1 -1| 2+ρ ]a 2+ρ n /(v n + na 2 n σ 2 ) (2+ρ)/2 .

,(7.19)

with the convention √ v 0 /0a 0 = 1. By (4.1) and (4.2), and since p ∈ (0, 3/4), we have

where

t r-3 k,n a 3 k .

By (4.1), it holds

By (4.1) and (4.2), when p ∈ (0, 3/4), v

n . Therefore, for any p ∈ (0, 3/4),

We handle now the term B n in (9.9). By taking into account (7.12), we derive

When, p in (0, 3/4), we have

n n -1/2 (log n). Therefore, for any p in (0, 3/4),

So, starting from (9.9) and taking into account (9.10) and (9.11), we derive that, for any r in (0, 2] and any p in (0, 3/4),

Thus, by taking into account (7.12), we have for p ∈ (0, 3/4) and r ∈ (0, 2],

11 Proof of Theorem 3.6

We start again from the decomposition (6.1) of Section 6 and use the notations of Section 10. By the triangle inequality

We first deal with the first term on the right hand side of (11.1). By Fact 1.1 in [START_REF] Dedecker | Quadratic transportation cost in the conditional central limit theorem for dependent sequences[END_REF], note that

Now, since V n is F-measurable and G 1,n is independent of F, we get W r (P Un+Vn|F , P G1,n+Vn|F ) = W r (P Un|F , P G1,n ) a.s. (11.3) Conditionally on F, U n is a sum of independent random variables with variance a 2 n σ 2 /(v n + na 2 n σ 2 ) and absolute moment of order 2 + ρ bounded by

Applying Corollary 1.2 in [START_REF] Bobkov | Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances[END_REF], we get for any r ∈ [1, 2] and ρ ∈ (0, r],

W r (P

where C is a constant depending only on r, σ 2 and EZ 2+ρ 1 . From (11.2) and (11.3), we get

(11.4)

We now deal with the second term on the right hand side of (11.1). Since G 1,n is independent of (V n , G 2,n ), it is easy to see that

By using inequality (3.2), we get that The result follows by considering (11.1) together with (11.4) and (11.5).