Xiequan Fan
email: fanxiequan@hotmail.com

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

Qi-Man Shao
email: ‡qmshao@sta.cuhk.edu.hk

Self-normalized Cramér type moderate deviations for martingales

Keywords: Martingales, self-normalized sequences, Cramér's moderate deviations

Let (Xi, Fi) i≥1 be a sequence of martingale differences. Set Sn = n i=1 Xi and

Introduction

Let (X i ) i≥1 be a sequence of independent random variables with zero means and finite variances: EX i = 0 and 0 < EX 2 i < ∞ for all i ≥ 1. Set

S n = n i=1 X i , B 2 n = n i=1 EX 2 i , V 2 n = n i=1 X 2 i .
It is well-known that under the Lindeberg condition the central limit theorem (CLT) holds sup

x∈R P(S n /B n ≤ x) -Φ(x) → 0 as n → ∞,
where Φ(x) denotes the standard normal distribution function. Cramér's moderate deviation expansion stated below gives an estimation of the relative error of P(S n /B n ≥ x) to 1 -Φ(x). If (X i ) i≥1 are identically distributed with Ee t0 √ |X1| < ∞ for some t 0 > 0 (cf. [START_REF] Linnik | On the probability of large deviations for the sums of independent variables[END_REF]), then for 0 ≤ x = o(n 1/6 ) as n → ∞,

P(S n /B n ≥ x)
1 -Φ (x) = 1 + o(1) and P(S n /B n ≤ -x) Φ (-x) = 1 + o(1).

(1.1) However, the limit theorems for self-normalized partial sums of independent random variables have put a new countenance on the classical limit theorems. The study of selfnormalized partial sums S n /V n originates from Student's t-statistic. Student's t-statistic T n is defined by

T n = √ n X n / σ,
where

X n = S n n and σ 2 = n i=1 (X i -X n ) 2 n -1 .
It is known that for all x ≥ 0,

P T n ≥ x = P S n /V n ≥ x n n + x 2 -1 1/2
, see [START_REF] Chung | The approximate distribution of Student's statistic[END_REF]. So, if we get an asymptotic bound on the tail probabilities for selfnormalized partial sums, then we have an asymptotic bound on the tail probabilities for T n . [START_REF] Giné | When is the Student t-statistic asymptotically standard normal?[END_REF] gave a necessary and sufficient condition for the asymptotic normality. [START_REF] Slavova | On the Berry-Esseen bound for Student's statistic[END_REF] and [START_REF] Bentkus | A Berry-Esséen bound for Student's statistic in the non-i.i.d. case[END_REF] (see also [Bentkus and Götze, 1996]) obtained the Berry-Esseen bounds for self-normalized partial sums. See also [START_REF] Novak | Extreme value methods with applications to finance[END_REF] and [START_REF] Shao | Self-normalized limit theorems: A survey[END_REF] for Berry-Esseen type inequalities with explicit constants. [START_REF] Shao | Self-normalized large deviations[END_REF] established a self-normalized Cramér-Chernoff large deviation without any moment assumptions and [START_REF] Shao | A Cramér type large deviation result for Student's t-statistic[END_REF] proved a self-normalized Cramér moderate deviation theorem under (2 + ρ)th moments: if (X i ) i≥1 are independent and identically distributed with E|X 1 | 2+ρ < ∞, ρ ∈ (0, 1], then for 0 ≤ x = o(n ρ/(4+2ρ) ) as n → ∞, P(S n /V n ≥ x)

1 -Φ (x) = 1 + o(1).

(1.

2)

The expansion (1.2) was further extended to independent but not necessarily identically distributed random variables by [START_REF] Jing | Self-normalized Cramér-type large deviations for independent random variables[END_REF] under finite (2 + ρ)th moments, ρ ∈ (0, 1], showing that

P(S n /V n ≥ x) 1 -Φ (x) = exp O 1 (1 + x) 2+ρ ι ρ n (1.3) uniformly for 0 ≤ x = o(min{ι -1 n , ς -1 n }),
where O(1) is bounded by an absolute constant and

ι ρ n = n i=1 E|X i | 2+ρ /B 2+ρ n and ς 2 n = max 1≤i≤n EX 2 i /B 2 n .
(1.4)

For further self-normalized Cramér type moderate deviation results for independent random variables we refer, for example, to [START_REF] Hu | Cramér type moderate deviations for the maximum of self-normalized sums[END_REF], [START_REF] Liu | Self-normalized Cramér type moderate deviations imsart-bj ver[END_REF], and [START_REF] Shao | Cramér type moderate deviation theorems for selfnormalized processes[END_REF]. We also refer to [de la [START_REF] De La Peña | Self-normalized Processes: Theory and Statistical Applications[END_REF] and [START_REF] Shao | Self-normalized limit theorems: A survey[END_REF] for recent developments in this area.

imsart-bj ver. 2014/10/16 file: Cramer_type_large_deviations_for_martingales-180621.tex date: July 26, 2019

The theory for self-normalized sums of independent random variables has been studied in depth. However, we are not aware of any such results for martingales. For some closely related topic, that is, exponential inequalities for self-normalized martingales, we refer to [de la [START_REF] De La Peña | A general class of exponential inequalities for martingales and ratios[END_REF], [START_REF] Bercu | Exponential inequalities for self-normalized martingales with applications[END_REF], [START_REF] Chen | Exponential inequalities for self-normalized martingales[END_REF] and [START_REF] Bercu | Concentration inequalities for sums and martingales[END_REF]. The main purpose of this paper is to establish selfnormalized Cramér type moderate deviation results for martingales. Let (δ n ) n≥1 , (ε n ) n≥1 and (κ n ) n≥1 be three sequences of nonnegative numbers, such that δ n → 0, ε n → 0 and κ n → 0 as n → ∞. Let (X i , F i ) i≥1 be a sequence of martingale differences satisfying

n i=1 E[X 2 i |F i-1 ] -B 2 n ≤ δ 2 n B 2 n , n i=1 E[|X i | 2+ρ |F i-1 ] ≤ ε ρ n B 2+ρ n ,
and max

1≤i≤n E[X 2 i |F i-1 ] ≤ κ 2 n B 2 n ,
where ρ ∈ (0, 3 2 ]. Here and hereafter, the inequalities between random variables are understood in the P-almost sure sense. From Corollary 2.1 we have

P(S n /V n ≥ x) = (1 -Φ(x))(1 + o(1)) (1.5) uniformly for 0 ≤ x = o( min{ε -ρ/(3+ρ) n , δ -1 n , κ -1 n }) as n → ∞.
A more general Cramér type expansion is obtained in a larger range in our Theorem 2.1, from which we derive a moderate deviation principle for self-normalized martingales. Moreover, when the condition

n i=1 E[|X i | 2+ρ |F i-1 ] ≤ ε ρ n B 2+ρ
n is replaced by a slightly stronger condition

E[|X i | 2+ρ |F i-1 ] ≤ (ε n B n ) ρ E[X 2 i |F i-1 ],
equality (1.5) holds for a larger range of 0 ≤ x = o( min{ε -ρ/(4+2ρ) n

, δ -1 n }) for ρ ∈ (0, 1], see Corollary 2.4. Clearly, our results recover (1.2) for i.i.d. random variables.

The rest of the paper is organized as follows. Our main results are stated and discussed in Section 2. Section 3 provides the preliminary lemmas that are used in the proofs of the main results. In Section 4, we prove the main results.

Throughout the paper the symbols c and c α , probably supplied with some indices, denote respectively a generic positive absolute constant and a generic positive constant depending only on α. Moreover, θ stands for values satisfying |θ| ≤ 1.

Main results

Let (X i , F i ) i=0,...,n be a sequence of martingale differences defined on a probability space (Ω, F, P), where X 0 = 0 and {∅, Ω} = F 0 ⊆ ... ⊆ F n ⊆ F are increasing σ-fields. Set

S 0 = 0, S k = k i=1 X i , k = 1, ..., n.
(2.1)

Then S = (S k , F k ) k=0,...,n is a martingale. Denote B 2 n = n i=1 EX 2 i . Let [S]
and S be, respectively, the square bracket and the conditional variance of the martingale S, that is

[S] 0 = 0, [S] k = k i=1 X 2 i , k = 1, ..., n,
and

S 0 = 0, S k = k i=1 E[X 2 i |F i-1 ], k = 1, ..., n. (2.2)
In the sequel, we use the following conditions:

(A1) There exists δ n ∈ [0, 1 4 ] such that n i=1 E[X 2 i |F i-1 ] -B 2 n ≤ δ 2 n B 2 n ;
(A2) There exist ρ > 0 and

ε n ∈ (0, 1 4 ] such that n i=1 E[|X i | 2+ρ |F i-1 ] ≤ ε ρ n B 2+ρ n ; (A3) There exists κ n ∈ (0, 1 4 ] such that E[X 2 i |F i-1 ] ≤ κ 2 n B 2 n , 1 ≤ i ≤ n;
(A4) There exist ρ ∈ (0, 1] and γ n ∈ (0, 1 4 ] such that

E[|X i | 2+ρ |F i-1 ] ≤ (γ n B n ) ρ E[X 2 i |F i-1 ], 1 ≤ i ≤ n.
When ρ ∈ (0, 1] and γ n ≤ (16/17) 1/ρ /4, conditions (A1) and (A4) imply condition (A2) with ε n = (17/16) 1/ρ γ n . Thus, we may assume that ε n = O(γ n ) as n → ∞. It is also easy to see that condition (A4) implies condition (A3) with κ n = γ n , see Lemma 3.5.

In practice, we usually have max{δ n , ε n , γ n , κ n } → 0 as n → ∞. In the case of sums of i.i.d. random variables, conditions (A1), (A2), (A3), and (A4) are satisfied with

δ n = 0, ε n , γ n , κ n = O( 1 √ n ).
Our first main result is the following Cramér type moderate deviation for the selfnormalized martingale

W n = S n / [S] n ,
under conditions (A1), (A2), and (A3).

Theorem 2.1. Assume that conditions (A1), (A2), and (A3) are satisfied. Set Then for 0

ρ 1 = min{ρ, 1}.
≤ x = o(min{ε -1 n , κ -1 n }), P(W n ≥ x) 1 -Φ (x) = exp θc ρ x 2+ρ1 ε ρ1 n + x 2 δ 2 n + (1 + x) ε ρ/(3+ρ) n + δ n . (2.3)
Under condition (A2) the best Berry-Esseen bound for standardized martingales is provided by [START_REF] Haeusler | On the rate of convergence in the central limit theorem for martingales with discrete and continuous time[END_REF]: assuming S n = B 2 n a.s., Haeusler proved that

sup x P(S n /B n ≤ x) -Φ (x) ≤ c ρ n i=1 E|X i /B n | 2+ρ 1/(3+ρ) .
Moreover, it was showed that this bound cannot be improved for martingales with finite (2 + ρ)th moments. In fact, there exist a positive constant c 0,ρ and a sequence of mar- 3+ρ) for all large enough n. In particular, under conditions (A2) and S n = B 2 n a.s., Haeusler's result implies that

tingale differences satisfying P(S n ≤ 0) -Φ (0) ≥ c 0,ρ n i=1 E|X i /B n | 2+ρ 1/(
sup x P(S n /B n ≤ x) -Φ (x) ≤ c ρ ε ρ/(3+ρ) n .
(2.4)

Notice that Theorem 2.1 implies that, for each absolute constant c > 0 there is a positive constant c ρ depending on ρ such that for n large enough,

sup |x|≤c P(W n ≤ x) -Φ (x) ≤ c ρ ε ρ/(3+ρ) n + δ n .
(2.5)

Under conditions (A2) and S n = B 2 n a.s., the bound in (2.5) for self-normalized martingales is of the same order as the bound in (2.4) for standardized martingales.

From Theorem 2.1, we obtain the following result about the equivalence to the normal tail.

Corollary 2.1. Assume that conditions (A1), (A2), and (A3) are satisfied with ρ ∈ (0, 3 2 ]. Then

P(W n ≥ x) 1 -Φ (x) = 1 + o(1) uniformly for 0 ≤ x = o( min{ε -ρ/(3+ρ) n , κ -1 n , δ -1 n }) as n → ∞.
Theorem 2.1 also implies the following moderate deviation principles (MDP) for selfnormalized martingales.

Corollary 2.2. Assume conditions (A1), (A2), and (A3) with max{δ n , ε n , κ n } → 0 as n → ∞. Let a n be any sequence of real numbers satisfying a n → ∞ and a n ε n → 0 as n → ∞. Then for each Borel set B,

-inf x∈B o x 2 2 ≤ lim inf n→∞ 1 a 2 n ln P W n a n ∈ B ≤ lim sup n→∞ 1 a 2 n ln P W n a n ∈ B ≤ -inf x∈B x 2 2 , (2.6)
where B o and B denote the interior and the closure of B, respectively.

The last corollary shows that the convergence speed of MDP depends only on ε n and it has nothing to do with the convergence speeds of κ n and δ n .

For i.i.d. random variables, the self-normalized MDP was established by [START_REF] Shao | Self-normalized large deviations[END_REF]. See also [START_REF] Jing | Self-normalized moderate deviations for independnet random variables[END_REF] for non-identically distributed random variables.

The other main results concern some improvements of Theorem 2.1 when condition (A3) is replaced by the stronger condition (A4). Theorems 2.2 and 2.3 below give respectively lower and upper bounds, while Theorem 2.4 gives a Cramér type moderate deviation expansion sharper than that in Theorem 2.1.

Theorem 2.2. Assume that conditions (A1), (A2), and (A4) are satisfied.

[i] If ρ ∈ (0, 1), then for 0 ≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) ≥ exp -c ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x) (x ρ γ ρ n + γ ρ n + δ n ) .
(2.7)

[ii] If ρ = 1, then for 0 ≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) ≥ exp -c x 3 ε n + x 2 δ 2 n + (1 + x) (xγ n + γ n | ln γ n | + δ n ) . (2.8)
The term γ n | ln γ n | in (2.8) cannot be replaced by γ n under the stated conditions. Indeed, [START_REF] Bolthausen | Exact convergence rates in some martingale central limit theorems[END_REF] (see Example 2 therein) showed that there exists a sequence of martingale differences satisfying |X i | ≤ 2 and S n = n a.s., such that for all n large enough,

P(S n ≥ 0) -Φ (0) ≥ c log n √ n , (2.9)
where c does not depend on n. Inequality (2.9) shows that the term γ n | ln γ n | in (2.8) cannot be replaced by γ n even for bounded martingale differences.

For any sequence of positive numbers (α n ) n≥1 denote

α n (x, ρ) = α ρ(2-ρ)/4 n 1 + x ρ(2+ρ)/4 .
(2.10) Accordingly, we shall use below the notations ε n (x, ρ) and γ n (x, ρ), which mean sequences defined by (2.10) with α n replaced by ε n and γ n .

Theorem 2.3. Assume that conditions (A1), (A2), and (A4) are satisfied.

[i] If ρ ∈ (0, 1), then for 0

≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) ≤ exp c ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x) x ρ γ ρ n + γ ρ n + δ n + ε n (x, ρ) . [ii] If ρ = 1, then for 0 ≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) ≤ exp c x 3 ε n + x 2 δ 2 n + (1 + x) xγ n + γ n | ln γ n | + δ n + ε n (x, 1
) .

Combining Theorems 2.2 and 2.3, we obtain the following Cramér type moderate deviation expansion for self-normalized martingales under conditions (A1), (A2), and (A4), which is stronger than the expansion in Theorem 2.1 since the term ε ρ/(3+ρ) n therein is improved to a smaller one.

Theorem 2.4. Assume that conditions (A1), (A2), and (A4) are satisfied.

[i] If ρ ∈ (0, 1), then for 0 ≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) = exp θc ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x) x ρ γ ρ n + γ ρ n + δ n + ε n (x, ρ) . [ii] If ρ = 1, then for 0 ≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) = exp θc x 3 ε n + x 2 δ 2 n + (1 + x) xγ n + γ n | ln γ n | + δ n + ε n (x, 1
) .

Notice that condition (A4) implies condition (A2) with ε n = γ n . Therefore, it follows from Theorem 2.4 that:

Corollary 2.3. Assume that conditions (A1) and (A4) are satisfied.

[i] If ρ ∈ (0, 1), then for 0 ≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) = exp θc ρ x 2+ρ γ ρ n + x 2 δ 2 n + (1 + x) δ n + γ n (x, ρ) . [ii] If ρ = 1, then for 0 ≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) = exp θc x 3 γ n + x 2 δ 2 n + (1 + x) δ n + γ n | ln γ n | + γ n (x, 1
) .

From Theorem 2.4, we also obtain the following result about the equivalence to the normal tail.

X. Fan et al.

Corollary 2.4. Assume conditions (A1), (A2), and (A4) with ρ ∈ (0, 1]. Then

P(W n ≥ x) 1 -Φ (x) = 1 + o(1) (2.11) uniformly for 0 ≤ x = o( min{ε -ρ/(2+ρ) n , γ -ρ/(1+ρ) n , δ -1 n }) as n → ∞.
In the case of i.i.d. random variables, conditions (A1), (A2), and (A4) are satisfied with

ε n , γ n = O(1/ √ n ) and δ n = 0. Thus, the range 0 ≤ x = o( min{ε -ρ/(2+ρ) n , δ -1 n , γ -ρ/(1+ρ) n }) reduces to 0 ≤ x = o(n -ρ/(4+2ρ) ), n → ∞,
which is the best possible result such that (2.11) holds (see [START_REF] Shao | A Cramér type large deviation result for Student's t-statistic[END_REF]). Moreover, from Theorem 2.4, we can get the estimation of the rate of convergence in (2.11); for example, when ρ = 1 we have:

Corollary 2.5. Assume conditions (A1), (A2), and (A4)

with ρ = 1, ε n , γ n , δ n = O(1/ √ n ). Then with c 0 > 0 for c 0 n 3/22 ≤ x = o(n 1/2 ) as n → ∞, P(W n ≥ x) 1 -Φ (x) = exp θc x 3 n 1/2 .
(2.12)

In particular, with c 0 , c 1 > 0 for c 0 n 3/22 ≤ x ≤ c 1 n 1/6 , P(W n ≥ x) 1 -Φ (x) -1 ≤ c x 3 n 1/2 . (2.13)
Notice that the rate of convergence in (2.12) coincides with that in (1.3) for i.i.d. random variables.

Remark 2.1. Notice that if (S k , F k ) k=0,...,n satisfies conditions (A1), (A2), (A3), and (A4), then (-S k , F k ) k=0,...,n also satisfies the same conditions. Thus the assertions in Theorems 2.1-2.4 and Corollaries 2.1-2.5 remain valid when P(Wn≥x)

1-Φ(x)
is replaced by

P(Wn≤-x) Φ(-x)
.

Preliminary lemmas

The proofs of Theorems 2.1-2.4 are based on a conjugate multiplicative martingale technique for changing the probability measure which is similar to that of the transformation of [START_REF] Esscher | On a method of determining correlation from the ranks of the variates[END_REF]. Our approach is inspired by the earlier work of [START_REF] Grama | Large deviations for martingales via Cramér's method[END_REF] on Cramér moderate deviations for standardized martingales, and by that of [START_REF] Shao | A Cramér type large deviation result for Student's t-statistic[END_REF], [START_REF] Jing | Self-normalized Cramér-type large deviations for independent random variables[END_REF], who developed techniques for moderate deviations of self-normalized sums of independent random variables. We extend these work by introducing a new choice of the density for the change of measure and refining the approaches in [START_REF] Shao | A Cramér type large deviation result for Student's t-statistic[END_REF] and [START_REF] Jing | Self-normalized Cramér-type large deviations for independent random variables[END_REF] to handle self-normalized martingales.

A key point of the proof is a new Berry-Esseen bound for martingales under the changed measure, see Proposition 3.1 below. Let

ξ i = X i B n , i = 1, ..., n.
Then (ξ i , F i ) i=0,...,n is also a sequence of martingale differences. Moreover, for simplicity of notations, set

M k = k i=1 ξ i , [M ] k = k i=1 ξ 2 i and M k = k i=1 E[ξ 2 i |F i-1 ], k = 1, ..., n. Thus W n = S n [S] n = M n [M ] n . (3.1)
For any real number λ, consider the exponential multiplicative martingale

Z(λ) = (Z k (λ), F k ) k=0,...,n , where Z 0 (λ) = 1, Z k (λ) = k i=1 e ζi(λ) E[e ζi(λ) |F i-1 ] , k = 1, ..., n with ζ i (λ) = λξ i -λ 2 ξ 2 i /2
. Thus, for each real number λ and each k = 1, ..., n, the random variable Z k (λ) is nonnegative and EZ k (λ) = 1. The last observation allows us to introduce the conjugate probability measure P λ := P λ,n on (Ω, F) defined by

dP λ = Z n (λ)dP. (3.2) Although (M k , F k ) k=0,.
..,n is a martingale under the measure P, it is no longer a martingale under the conjugate probability measure P λ . To obtain a martingale under P λ we have to center the random variables ζ i (λ). Denote by E λ the expectation with respect to P λ . Because Z(λ) is a uniformly integrable martingale under P, we have

E λ [ζ] = E[ζZ n (λ)] (3.3) and E λ [ζ|F i-1 ] = E[ζe ζi(λ) |F i-1 ] E[e ζi(λ) |F i-1 ] (3.4) for any F i -measurable random variable ζ that is integrable with respect to F i . Set b i (λ) = E λ [ζ i (λ)|F i-1 ], i = 1, . . . ,
η i (λ) = ζ i (λ) -b i (λ), i = 1, . . . , n,
and

Y k (λ) = k i=1 η i (λ), k = 1, ..., n. (3.5) Then Y (λ) = (Y k (λ), F k ) k=0,.
..,n is the conjugate martingale. The following semimartingale decomposition is well-known:

k i=1 ζ i (λ) = B k (λ) + Y k (λ), k = 1, ..., n, (3.6)
where

B(λ) = (B k (λ), F k ) k=0,.
..,n is the drift process defined as

B k (λ) = k i=1 b i (λ), k = 1, ..., n.
By the relation between E and E λ on F i , we have

b i (λ) = E[ζ i (λ)e ζi(λ) |F i-1 ] E[e ζi(λ) |F i-1 ] , i = 1, ..., n. (3.7)
It is easy to compute the conditional variance of the conjugate martingale Y (λ) under the measure P λ , for k = 0, ..., n,

Y (λ) k = k i=1 E λ [η i (λ) 2 |F i-1 ] = k i=1 E λ [(ζ i (λ) -b i (λ)) 2 |F i-1 ] = k i=1 E[ζ 2 i (λ)e ζi(λ) |F i-1 ] E[e ζi(λ) |F i-1 ] - E[ζ i (λ)e ζi(λ) |F i-1 ] 2 E[e ζi(λ) |F i-1 ] 2 . (3.8)
In the sequel, we give the upper and lower bounds for B n (λ). To this end, we need the following three useful lemmas. Their proofs are not given here but they are similar to those of the corresponding assertions in [START_REF] Shao | A Cramér type large deviation result for Student's t-statistic[END_REF] and [START_REF] Jing | Self-normalized Cramér-type large deviations for independent random variables[END_REF] established for independent random variables. Set

ε i,λ = λ 2 E[ξ 2 i 1 {|λξi|>1} |F i-1 ] + λ 3 E[|ξ i | 3 1 {|λξi|≤1} |F i-1 ], λ ≥ 0. If E[|ξ i | 2+ρ ] < ∞ for ρ ∈ [0, 1], then it is obvious that ε i,λ ≤ λ 2+ρ E[|ξ i | 2+ρ |F i-1 ], λ ≥ 0.
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Lemma 3.1. For all λ > 0 and τ ∈ [ 1 8 , 2], we have

E[e λξi-τ λ 2 ξ 2 i |F i-1 ] -1 -( 1 2 -τ )λ 2 E[ξ 2 i |F i-1 ] ≤ c ε i,λ .
Lemma 3.2. For all λ > 0, we have

E[e ζi(λ) |F i-1 ] -1 ≤ c ε i,λ , E[ζ i (λ)e ζi(λ) |F i-1 ] - 1 2 λ 2 E[ξ 2 i |F i-1 ] ≤ c ε i,λ , E[ζ 2 i (λ)e ζi(λ) |F i-1 ] -λ 2 E[ξ 2 i |F i-1 ] ≤ c ε i,λ , E[|ζ i (λ)| 3 e ζi(λ) |F i-1 ] ≤ c ε i,λ , E[ζ i (λ)e ζi(λ) |F i-1 ] 2 ≤ c ε i,λ . Lemma 3.3. Let H i = ξ 2 i -E[ξ 2 i |F i-1 ]. Then for all λ > 0, E[H i e ζi(λ) |F i-1 ] ≤ c 1 λ 2 ε i,λ .
Using Lemma 3.2, we obtain the following upper and lower bounds for B n (λ).

Lemma 3.4. Assume conditions (A2) and (A3) with ρ ∈ (0, 1]. Then for

0 ≤ λ = o(max{ε -1 n , κ -1 n }), B n (λ) - 1 2 λ 2 M n ≤ c λ 2+ρ ε ρ n .
(3.9)

Proof. According to the definition of b i (λ), we have

b i (λ) = E[ζ i (λ)e ζi(λ) |F i-1 ] E[e ζi(λ) |F i-1 ] .
By Lemma 3.2, it follows that

E[ζ i (λ)e ζi(λ) |F i-1 ] - 1 2 λ 2 E[ξ 2 i |F i-1 ] ≤ c ε i,λ
and

E[e ζi(λ) |F i-1 ] -1 ≤ c ε i,λ . (3.10)
Therefore, conditions (A2) and (A3) imply that for 0

≤ λ = o(max{ε -1 n , κ -1 n }), b i (λ) - 1 2 λ 2 E[ξ 2 i |F i-1 ] ≤ c ε i,λ and B n (λ) - 1 2 λ 2 M n ≤ c λ 2+ρ ε ρ n as desired.
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X. Fan et al.

The following lemma shows that condition (A4) implies condition (A3) with κ n = γ n .

Lemma 3.5. Assume condition (A4). Then

E[ξ 2 i |F i-1 ] ≤ γ 2 n .
Proof. By Jensen's inequality and condition (A4), it holds that

E[ξ 2 i |F i-1 ] (2+ρ)/2 ≤ E[|ξ i | 2+ρ |F i-1 ] ≤ γ ρ n E[ξ 2 i |F i-1 ], from which we get E[ξ 2 i |F i-1 ] ≤ γ 2 n .
Lemma 3.6. Assume condition (A4). Then for any t ∈ [0, ρ),

E[|ξ i | 2+t |F i-1 ] ≤ γ t n E[ξ 2 i |F i-1 ]. (3.11)
Proof. Let l, p, q be defined by the following equations lp = 2, (2 + t -l)q = 2 + ρ, p -1 + q -1 = 1, l > 0, and p, q ≥ 1.

Solving the last equations, we get

l = 2(ρ -t) ρ , p = ρ ρ -t , q = ρ t .
By Hölder's inequality and condition (A4), it is easy to see that

E[|ξ i | 2+t |F i-1 ] = E[|ξ i | l |ξ i | 2+t-l |F i-1 ] ≤ (E[|ξ i | lp |F i-1 ]) 1/p (E[|ξ i | (2+t-l)q |F i-1 ]) 1/q ≤ (E[ξ 2 i |F i-1 ]) 1/p (E[|ξ i | 2+ρ |F i-1 ]) 1/q ≤ (E[ξ 2 i |F i-1 ]) 1/p (γ ρ n E[ξ 2 i |F i-1 ]) 1/q ≤ γ ρ/q n E[ξ 2 i |F i-1 ] = γ t n E[ξ 2 i |F i-1 ].
This completes the proof of the lemma.

Lemma 3.7. Assume conditions (A1) and (A2). Then for any t ∈ [0, ρ),

n i=1 E[|ξ i | 2+t |F i-1 ] ≤ 2 ε t n .
(3.12)

Proof. Recall the notations in the proof of Lemma 3.6. It is easy to see that

n i=1 E[|ξ i | 2+t |F i-1 ] ≤ n i=1 (E[ξ 2 i |F i-1 ]) 1/p (E[|ξ i | 2+ρ |F i-1 ]) 1/q .
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Using Hölder's inequality and conditions (A1) and (A2), we have

n i=1 E[|ξ i | 2+t |F i-1 ] ≤ n i=1 E[ξ 2 i |F i-1 ] 1/p n i=1 E[|ξ i | 2+ρ |F i-1 ] 1/q ≤ 2 ε t n ,
which gives the desired inequality.

We will also need the following two lemmas.

Lemma 3.8. Assume condition (A1). Then for all x > 0,

P M n ≥ x [M ] n , [M ] n ≥ 16 ≤ 2 3 x -2/3 exp - 3 4 x 2 .
Proof. By inequality (11) of [START_REF] Delyon | Exponential inequalities for sums of weakly dependent variables[END_REF], we have for all λ ∈ R,

E exp λM n - λ 2 2 ( 1 3 [M ] n + 2 3 M n ) ≤ 1.
Applying the last inequality to the exponential inequality of [de la Peña and Pang, 2009] with p = q = 2, we deduce that for all x > 0,

P |M n | 3 2 ( 1 3 [M ] n + 2 3 M n + EM 2 n ) ≥ x ≤ 2 3 2/3 x -2/3 exp - 1 2 x 2 . (3.13)
By condition (A1) and the fact that

E M n = EM 2 n = 1, it is easy to see that 3 2 M n + 9 4 EM 2 n ≤ 3 2 (1 + δ 2 n ) + 9 4 ≤ 3 2 1 + 1 16 + 9 4 < 4.
Therefore, for all x > 0,

P M n ≥ x [M ] n , [M ] n ≥ 16 ≤ P M n ≥ x 3 4 [M ] n + 4, [M ] n ≥ 16 ≤ P M n ≥ x 3 4 [M ] n + 3 2 M n + 9 4 EM 2 n , [M ] n ≥ 16 ≤ P M n ≥ x 3 4 [M ] n + 3 2 M n + 9 4 EM 2 n = P M n ≥ 3 2 x 1 2 [M ] n + M n + 3 2 EM 2 n ≤ 2 3 x -2/3 exp - 3 4 x 2
as desired.

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Lemma 3.9. Assume conditions (A1) and (A2). Then for all ρ > 0,

P |[M ] n -M n | ≥ 1 ≤ c ρ ε (2+ρ)/2 n + ε ρ n . Proof. Notice that [M ] n -M n = n i=1 (ξ 2 i -E[ξ 2 i |F i-1 ]
) is a martingale. For ρ, we distinguish two cases as follows.

When ρ ∈ (0, 2], by the inequality of [START_REF] Von Bahr | Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2[END_REF], it follows that

E[|[M ] n -M n | (2+ρ)/2 ] ≤ n i=1 E[|ξ 2 i -E[ξ 2 i |F i-1 ]| (2+ρ)/2 ] ≤ c 1 n i=1 E[|ξ i | 2+ρ ] ≤ c 2 ε ρ n ,
where the last line follows by conditions (A1) and (A2). Hence, by Markov's inequality,

P |[M ] n -M n | ≥ 1 ≤ E[|[M ] n -M n | (2+ρ)/2 ] ≤ c 2 ε ρ n ,
When ρ > 2, by Rosenthal's inequality (cf., Theorem 2.12 of [START_REF] Hall | Martingale Limit Theory and its Applications[END_REF]), Lemma and condition (A2), it follows that

E[|[M ] n -M n | (2+ρ)/2 ] ≤ c ρ,1 E n i=1 E[(ξ 2 i -E[ξ 2 i |F i-1 ]) 2 |F i-1 ] (2+ρ)/4 + n i=1 E|ξ 2 i -E[ξ 2 i |F i-1 ]| (2+ρ)/2 ≤ c ρ,2 E n i=1 E[ξ 4 i |F i-1 ] (2+ρ)/4 + n i=1 E|ξ i | 2+ρ ≤ c ρ,3 ε (2+ρ)/2 n + ε ρ n . (3.14)
This completes the proof of the lemma.

Consider the predictable process Ψ(λ) = (Ψ k (λ), F k ) k=0,...,n , which is related to the martingale M as follows:

Ψ k (λ) = k i=1 ln E[e ζi(λ) |F i-1 ].
(3.15)

By equality (3.10), we easily obtain the following elementary bound for the process Ψ(λ).

Lemma 3.10. Assume conditions (A2) and (A3) with ρ ∈ (0, 1]. Then for 0

≤ λ = o(min{ε -1 n , κ -1 n }), Ψ n (λ) ≤ c λ 2+ρ ε ρ n .
In the proofs of Theorems 2.2 and 2.3, we make use of the following assertion, which gives us a rate of convergence in the CLT for the conjugate martingale Y (λ) under the probability measure P λ .

Proposition 3.1. Assume conditions (A1) and (A4). With the convention that Y n (0)/0 = M n , we have:

[i] If ρ ∈ (0, 1), then for 0 ≤ λ = o(γ -1 n ), sup x P λ (Y n (λ)/λ ≤ x) -Φ(x) ≤ c ρ λ ρ γ ρ n + γ ρ n + δ n . [ii] If ρ = 1, then for 0 ≤ λ = o(γ -1 n ), sup x P λ (Y n (λ)/λ ≤ x) -Φ(x) ≤ c λγ n + γ n | ln γ n | + δ n .
Similarly, we have the following Berry-Esseen bound.

Proposition 3.2. Assume conditions (A1), (A2) and (A3). Then for

0 ≤ λ = o(max{ε -1 n , κ -1 n }), sup x P λ (Y n (λ)/λ ≤ x) -Φ(x) ≤ c ρ λ ρ/2 γ ρ/2 n + ε ρ/(3+ρ) n + δ n ,
with the convention that Y n (0)/0 = M n .

The proofs of Propositions 3.1 and 3.2 are much more complicated and we give details in the supplemental article [START_REF] Fan | Supplement to "Self-normalized Cramér type moderate deviations for martingales[END_REF].

Proof of the main results

We start with the proofs of Theorems 2.2 and 2.3, and conclude with the proof of Theorem 2.1. Theorem 2.4 is an easy consequence of Theorems 2.2 and 2.3

4.1. Proof of Theorem 2.2 Recall that ζ i (λ) = λξ i - 1 2 λ 2 ξ 2 i .
By (3.1), it is easy to see that

S n ≥ x [S] n = M n ≥ x [M ] n ⊇ M n ≥ x 2 + λ 2 [M ] n 2λ = n i=1 ζ i (λ) ≥ x 2 2 .
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For 0 ≤ λ = o(γ -1 n ), according to (3.2), (3.6) and (3.15), we have the following representation:

P W n ≥ x = E λ Z n (λ) -1 1 {Sn≥x √ [S]n} = E λ exp - n i=1 ζ i (λ) + Ψ n (λ) 1 {Mn≥x √ [M ]n} ≥ E λ exp -Y n (λ) -B n (λ) + Ψ n (λ) 1 { n i=1 ζi(λ)≥ x 2 2 } = E λ exp -Y n (λ) -B n (λ) + Ψ n (λ) 1 {Yn(λ)≥ x 2
2 -Bn(λ)} . Using Lemmas 3.5, 3.4 and 3.10, we get

P W n ≥ x ≥ E λ exp -Y n (λ) - 1 2 λ 2 M n + c 1 λ 2+ρ ε ρ n ×1 {Yn(λ)≥ x 2 2 -( 1 2 λ 2 M n -c1λ 2+ρ ε ρ n )} . Condition (A1) implies that | M n -1| ≤ δ 2
n , and thus

P W n ≥ x ≥ E λ exp -Y n (λ) - 1 2 λ 2 + c 1 λ 2+ρ ε ρ n (1 + δ 2 n ) ×1 {Yn(λ)≥ x 2 2 -( 1 2 λ 2 (1-δ 2 n )-c1λ 2+ρ ε ρ n )} . (4.1)
Let λ = λ(x) be the largest solution of the following equation

1 2 λ 2 (1 -δ 2 n ) -c 1 λ 2+ρ ε ρ n = x 2 2 .
The definition of λ implies that for 0

≤ x = o(γ -1 n ), x ≤ λ ≤ c 2 x 1 -δ 2 n (4.2) and λ = x + c 3 θ 0 (x 1+ρ ε ρ n + xδ 2 n ), (4.3)
where 0 ≤ θ 0 ≤ 1. From (4.1), we obtain By integration by parts, we have the following bound:

P W n ≥ x ≥ exp - 1 2 λ 2 + c 1 λ 2+ρ ε ρ n (1 + δ 2 n ) E λ e -Yn(λ) 1 {Yn(λ)≥0} . (4.4) Setting F n (y) = P λ (Y n (λ) ≤ y), we get P W n ≥ x ≥ exp -c 4 λ 2 δ 2 n + λ 2+ρ ε ρ n - λ 2 2 ∞ 0 e -
∞ 0 e -y dF n (y) ≥ ∞ 0 e -y dΦ(y/λ) -2 sup y F n (y) -Φ(y/λ) . (4.6)
We distinguish two cases according to the values of ρ.

Case 1 : ρ ∈ (0, 1). Combining (4.5) and (4.6), by Proposition 3.1, we have for 0

≤ x = o(γ -1 n ), P W n ≥ x ≥ exp -c 4 λ 2 δ 2 n + λ 2+ρ ε ρ n - λ 2 2 × ∞ 0 e -λy dΦ(y) -c 1,ρ λ ρ γ ρ n + γ ρ n + δ n . (4.7) Because e -λ 2 /2 ∞ 0 e -λy dΦ(y) = 1 -Φ (λ) (4.8) and 1 1 + λ e -λ 2 /2 ≤ √ 2π 1 -Φ (λ) , λ ≥ 0, (4.9)
we obtain the following lower bound

P W n ≥ x 1 -Φ λ ≥ exp -c 4 (λ 2 δ 2 n + λ 2+ρ ε ρ n ) 1 -c 2,ρ (1 + λ)(λ ρ γ ρ n + γ ρ n + δ n ) ≥ exp -c 3,ρ λ 2 δ 2 n + λ 2+ρ ε ρ n + (1 + λ)(λ ρ γ ρ n + γ ρ n + δ n ) , (4.10) for 0 ≤ λ ≤ 1 2c2,ρ min{γ -ρ/(1+ρ) n , δ -1 n }. Next, we consider the case of 1 2c2,ρ min{γ -ρ/(1+ρ) n , δ -1 n } ≤ λ = o(γ -1 n ).
Let K ≥ 1 be an absolute constant, whose exact value is chosen later. It is easy to see that (4.11) where τ = λ ρ γ ρ n + δ n . By Proposition 3.1, we have

E λ e -Yn(λ) 1 {Yn(λ)≥0} ≥ E λ e -Yn(λ) 1 {0≤Yn(λ)≤λKτ } ≥ e -λKτ P λ 0 ≤ Y n (λ) ≤ λKτ ,
P λ 0 ≤ Y n (λ) ≤ λKτ ≥ P 0 ≤ N (0, 1) ≤ Kτ -c 4,ρ τ ≥ 1 √ 2π Kτ e -K 2 τ 2 /2 -c 4,ρ τ ≥ 1 3 K -c 4,ρ τ.
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Letting K ≥ 12c 4,ρ , it follows that

P λ 0 ≤ Y n (λ) ≤ λKτ ≥ 1 4 Kτ = 1 4 K λ 1+ρ γ ρ n + λδ n λ . Choosing K = max 12c 4,ρ , 4 √ π (2c 2,ρ ) 1+ρ
and taking into account that 1 2c2,ρ min{γ

-ρ/(1+ρ) n , δ -1 } ≤ λ = o(γ -1 n ), we conclude that P λ 0 ≤ Y n (λ) ≤ λKτ ≥ 1 √ πλ .
Because the inequality 1

√ πλ e -λ 2 /2 ≥ 1 -Φ (λ) is valid for all λ ≥ 1, it follows that for 1 2c2,ρ min{γ -ρ/(1+ρ) n , δ -1 } ≤ λ = o(γ -1 n ), P λ 0 ≤ Y n (λ) ≤ Kτ ≥ 1 -Φ λ e λ 2 /2 . (4.12)
Combining (4.4), (4.11), and (4.12), we obtain

P W n ≥ x 1 -Φ λ ≥ exp -c 5,ρ λ 2 δ 2 n + λ 2+ρ ε ρ n + (1 + λ)(λ ρ γ ρ n + γ ρ n + δ n ) , (4.13) which is valid for 1 2c2,ρ min{γ -ρ/(1+ρ) n , δ -1 } ≤ λ = o(γ -1 n ).
From (4.10) and (4.13), we get for 0 ≤ λ = o(γ -1 n ),

P W n ≥ x 1 -Φ λ ≥ exp -c 6,ρ λ 2 δ 2 n + λ 2+ρ ε ρ n + (1 + λ)(λ ρ γ ρ n + γ ρ n + δ n ) . (4.14)
Next, we substitute x for λ in the tail of the normal law 1 -Φ(λ). By (4.2), (4.3), and (4.9), we get

1 ≤ ∞ λ exp{-t 2 /2}dt ∞ x exp{-t 2 /2}dt ≤ 1 + x λ exp{-t 2 /2}dt ∞ x exp{-t 2 /2}dt ≤ 1 + c 1 x(x -λ) exp (x 2 -λ 2 )/2 ≤ exp c 2 (x 2 δ 2 n + x 2+ρ ε ρ n ) (4.15)
and hence

1 -Φ λ = (1 -Φ(x)) exp θ 1 c (x 2+ρ ε ρ n + x 2 δ 2 n ) . (4.16)
Implementing (4.16) in (4.14) and using (4.2), we obtain for 0

≤ x = o(γ -1 n ), P W n ≥ x 1 -Φ (x) ≥ exp -c 7,ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x)(x ρ γ ρ n + γ ρ n + δ n ) ,
which gives the desired lower bound (2.7).

Case 2 : ρ = 1. Using Proposition 3.1 with ρ = 1, we have for 0

≤ x = o(γ -1 n ), P W n ≥ x ≥ exp -c 1 λ 2 δ 2 n + λ 3 ε n - λ 2 2 × ∞ 0 e -λy dΦ(y) -c 2 λγ n + γ n | ln γ n | + δ n ,
that is, the term γ ρ n in inequality (4.7) has been replaced by γ n | ln γ n |. By an argument similar to that of Case 1, we obtain the desired lower bound (2.8).

Proof of Theorem 2.3

We first prove Theorem 2.3 for 1 ≤ x = o(γ -1 n ). Observe that

P W n ≥ x = P W n ≥ x, |[M ] n -M n | ≤ δ n + 1/(2x) + P W n ≥ x, |[M ] n -M n | > δ n + 1/(2x) . (4.17)
For the the first term on the right hand side of (4.17), by (3.2) and (3.5) with λ = x, we have the following representation:

P W n ≥ x, |[M ] n -M n | ≤ δ n + 1/(2x) = E x Z n (x) -1 1 {Mn≥x √ [M ]n, |[M ]n-M n |≤δn+1/(2x)} = E x e -Yn(x)-Bn(x)+Ψn(x) 1 xMn≥x 2 √ 1+[M ]n-1, |[M ]n-M n|≤δn +1/(2x)
.

By the inequality 1 + y ≥ 1 + y/2 -y 2 /2, y ≥ -1, condition (A1) and Lemma 3.4, we have for 1

≤ x = o(γ -1 n ), P W n ≥ x , |[M ] n -M n | ≤ δ n + 1/(2x) ≤ E x exp -Y n (x) -B n (x) + Ψ n (x) ×1 xMn-1 2 x 2 [M ]n+ 1 2 x 2 ([M ]n-1) 2 ≥ 1 2 x 2 , |[M ]n-M n |≤δn+1/(2x) ≤ E x exp -Y n (x) -B n (x) + Ψ n (x) ×1 xMn-1 2 x 2 [M ]n+x 2 ([M ]n-M n ) 2 +x 2 (1-M n) 2 ≥ 1 2 x 2 , |[M ]n-M n |≤δn+1/(2x) ≤ E x exp -Y n (x) -B n (x) + Ψ n (x) ×1 Yn(x)≥-x 2 ([M ]n-M n ) 2 -x 2 δ 4 n + 1 2 x 2 -Bn(x), |[M ]n-M n |≤δn+1/(2x)
.

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Thus, for 1 ≤ x = o(γ -1 n ), P W n ≥ x , |[M ] n -M n | ≤ δ n + 1/(2x) ≤ E x exp -Y n (x) -B n (x) + Ψ n (x) ×1 Yn(x)≥-x 2+ρ ε ρ n -x 2 δ 4 n + 1 2 x 2 -Bn(x), |[M ]n-M n|≤(xεn ) ρ/2 + E x exp -Y n (x) -B n (x) + Ψ n (x) ×1 0>Yn(x)≥-x 2 ([M ]n-M n ) 2 -x 2 δ 4 n + 1 2 x 2 -Bn(x), (xεn) ρ/2 <|[M ]n-M n|≤δn +1/(2x)
.

≤ I 1 (x) + I 2 (x), (4.18)
where

I 1 (x) = E x exp -Y n (x) -B n (x) + Ψ n (x) 1 Yn(x)≥-c1(x 2+ρ ε ρ n +x 2 δ 2 n )
and

I 2 (x) = E x exp -Y n (x) -B n (x) + Ψ n (x) ×1 0>Yn(x)≥-1-c2(x 2+ρ ε ρ n +x 2 δ 2 n ), (xεn) ρ/2 <|[M ]n-M n |≤δn+1/(2x)
.

For I 1 (x), by an argument similar to the proof of Theorem 2.2, we get for 1 ≤ x = o(γ -1 n ),

I 1 (x) 1 -Φ (x) ≤            exp c ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x) (x ρ γ ρ n + γ ρ n + δ n ) if ρ ∈ (0, 1), exp c x 3 ε n + x 2 δ 2 n + (1 + x) (xγ n + γ n | ln γ n | + δ n ) if ρ = 1. (4.19)
Next, consider the item I 2 (x). By condition (A1), Lemmas 3.4 and 3.10, it is obvious that for 1 Denote by

≤ x = o(γ -1 n ), I 2 (x) ≤ exp - 1 2 x 2 + c 1 x 2+ρ ε ρ n + x 2 δ 2 n ×E x e -Yn(x) 1 0>Yn(x)≥-1-c2(x 2+ρ ε ρ n +x 2 δ 2 n ), (xεn) ρ/2 <|[M ]n-M n | ≤ exp - 1 2 x 2 + c 1 x 2+ρ ε ρ n + x 2 δ 2 n ×E x e 1+c2(x 2+ρ ε ρ n +x 2 δ 2 n ) 1 (xεn) ρ/2 <|[M ]n-M n | ≤ exp 1 - 1 2 x 2 + c 3 x 2+ρ ε ρ n + x 2 δ 2 n E x 1 (xεn) ρ/2 <|[M ]n-M n | . (4.
M (x) n = n i=1 E x [ξ 2 i |F i-1 ]. Notice that ε n = O(γ n ).
From (3.4), using (3.10), Lemmas 3.3, 3.5 and condition (A2), we obtain for 1

≤ x = o(γ -1 n ), M (x) n -M n ≤ n i=1 E[ξ 2 i e xξi-x 2 ξ 2 i /2 |F i-1 ] E[e xξi-x 2 ξ 2 i /2 |F i-1 ] -E[ξ 2 i |F i-1 ] + n i=1 E[ξ i e xξi-x 2 ξ 2 i /2 |F i-1 ] 2 E[e xξi-x 2 ξ 2 i /2 |F i-1 ] 2 ≤ c 4 n i=1 E[x ρ |ξ i | 2+ρ |F i-1 ] + (E[xξ 2 i |F i-1 ]) 2 ≤ c 4 n i=1 E[x ρ |ξ i | 2+ρ |F i-1 ] + x 2 E[|ξ i | 2+ρ |F i-1 ](E[ξ 2 i |F i-1 ]) (2-ρ)/2 ≤ c 5 x ρ ε ρ n . (4.21) Thus, for 1 ≤ x = o(γ -1 n ), I 2 (x) ≤ exp 1 - 1 2 x 2 + c 3 x 2+ρ ε ρ n + x 2 δ 2 n E x 1 1 2 (xεn) ρ/2 <|[M ]n-M (x) n| ≤ 4e (xε n ) ρ(2+ρ)/4 exp - 1 2 x 2 + c 3 x 2+ρ ε ρ n + x 2 δ 2 n E x |[M ] n -M (x) n | (2+ρ)/2 .
It is obvious that

[M ] n -M (x) n = n i=1 (ξ 2 i -E x [ξ 2 i |F i-1 ]). Thus, ([M ] i -M (x) i , F i ) i=0,.
..,n is a martingale with respect to the probability measure P x . By the inequality of [START_REF] Von Bahr | Inequalities for the rth absolute moment of a sum of random variables, 1 ≤ r ≤ 2[END_REF], it follows that for 1

≤ x = o(γ -1 n ), E x [|[M ] n -M (x) n | (2+ρ)/2 ] ≤ c 6 n i=1 E x [|ξ 2 i -E x [ξ 2 i |F i-1 ]| (2+ρ)/2 ] ≤ c 7 n i=1 E x [|ξ i | 2+ρ ] = c 7 n i=1 E[|ξ i | 2+ρ e ζi(x) |F i-1 ] E[e ζi(x) |F i-1 ] ≤ c 8 ε ρ n . (4.22) Hence, for 1 ≤ x = o(γ -1 n ), I 2 (x) ≤ c ε ρ(2-ρ)/4 n x ρ(2+ρ)/4 exp - 1 2 x 2 + c 3 x 2+ρ ε ρ n + x 2 δ 2 n . ( 4
W n ≥ x, |[M ] n -M n | > δ n + 1/(2x) . Since |1 -M n | ≤ δ 2 n ≤ δ n /2, it is obvious that P W n ≥ x, |[M ] n -M n | > δ n + 1/(2x) ≤ P W n ≥ x, |[M ] n -1| + |1 -M n | > δ n + 1/(2x) ≤ P W n ≥ x, |[M ] n -1| > δ n /2 + 1/(2x) .
To estimate the tail probability in the last line, we follow the argument of [START_REF] Shao | Cramér type moderate deviation theorems for selfnormalized processes[END_REF]. We have the following decomposition:

P W n ≥ x, |[M ] n -1| > δ n /2 + 1/(2x) ≤ P M n / [M ] n ≥ x, 1 + δ n /2 + 1/(2x) < [M ] n ≤ 16 + P M n / [M ] n ≥ x, [M ] n < 1 -δ n /2 -1/(2x) + P M n / [M ] n ≥ x, [M ] n > 16 := 3 v=1 P (M n , [M ] n ) ∈ E v , (4.24)
where E v ⊂ R × R + , 1 ≤ v ≤ 3, are given by

E 1 = (u, v) ∈ R × R + : u/v ≥ x, 1 + δ n /2 + 1/(2x) < v ≤ 4 , E 2 = (u, v) ∈ R × R + : u/v ≥ x, v < 1 -δ n /2 -1/(2x) , E 3 = (u, v) ∈ R × R + : u/v ≥ x, v > 4 .
To estimate the probability P((M n , [M ] n ) ∈ E 1 ), we introduce the following new conjugate probability measure P x defined by

d P x = Z n (x)dP,
where

Z n (x) = k i=1 e ζi(x) E[e ζi(x) |F i-1 ] and ζ i (x) = xξ i -x 2 ξ 2 i /8.
Denote by E x the expectation with respect to P x and M (x

) n = n i=1 E x [ξ 2 i |F i-1 ]
. By an argument similar to (4.21), it follows that for 1

≤ x = o(γ -1 n ), M (x) n -M n ≤ c x ρ ε ρ n .
By Markov's inequality, we deduce that

P (M n , [M ] n ) ∈ E 1 ≤ (δ n /2 + 1/(2x)) -2 e -inf (u,v)∈E 1 (xu-(vx) 2 /8) E[([M ] n -1) 2 e xMn-[M ]nx 2 /8 ] ≤ 16x 2 e -inf (u,v)∈E 1 (xu-(vx) 2 /8) E[([M ] n -M (x) n ) 2 e xMn-[M ]nx 2 /8 ] + 16x 2 e -inf (u,v)∈E 1 (xu-(vx) 2 /8) E[( M (x) n -M n ) 2 e xMn-[M ]nx 2 /8 ] + 16δ -2 n e -inf (u,v)∈E 1 (xu-(vx) 2 /8) E[( M n -1) 2 e xMn-[M ]nx 2 /8 ] ≤ 16x 2 e -inf (u,v)∈E 1 (xu-(vx) 2 /8) E[([M ] n -M (x) n ) 2 e xMn-[M ]nx 2 /8 ] + c x 2+2ρ ε 2ρ n e -inf (u,v)∈E 1 (xu-(vx) 2 /8) E[e xMn-[M ]nx 2 /8 ] + 16δ 2 n e -inf (u,v)∈E 1 (xu-(vx) 2 /8) E[e xMn-[M ]nx 2 /8 ], (4.25)
where it is easy to verify that inf

(u,v)∈E1 xu - 1 8 (vx) 2 ≥ 7 8 x 2 + 1 4 x -c x 2 δ 2 n . (4.26)
By Lemma 3.1, conditions (A1) and (A2), it follows that where the last line follows because ([M ] i -M (x) i , F i ) i=0,...,n is a martingale with respect to the probability measure P x . Therefore, by Lemma 3.1, conditions (A1) and (A2) again, we have for 1

n i=1 E[e ζi(x) |F i-1 ] ≤ n i=1 1 + 3 8 x 2 E[ξ 2 i |F i-1 ] + c x 2+ρ E[|ξ i | 2+ρ |F i-1 ] ≤ n i=1 exp 3 8 x 2 E[ξ 2 i |F i-1 ] + c x 2+ρ E[|ξ i | 2+ρ |F i-1 ] = exp 3 8 x 2 M n + c x 2+ρ n i=1 E[|ξ i | 2+ρ |F i-1 ] ≤ exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) . Therefore, for 1 ≤ x = o(γ -1 n ), E ([M ] n -M (x) n ) 2 e xMn-[M ]nx 2 /8 = E Π n i=1 E[e ζi(x) |F i-1 ] [M ] n -M (x) n 2 Z n (x) ≤ E [M ] n -M (x) n 2 Z n (x) exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) = E x [M ] n -M (x) n 2 exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) = n i=1 E x (ξ 2 i -E x [ξ 2 i |F i-1 ]) 2 exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) , imsart-
≤ x = o(γ -1 n ), E ([M ] n -M (x) n ) 2 e xMn-[M ]nx 2 /8 ≤ n i=1 E x E x ξ 4 i |F i-1 ] exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) = n i=1 E x E[ξ 4 i e ζi(x) |F i-1 ] E[e ζi(x) |F i-1 ] exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) ≤ c 0 n i=1 E x 1 x 2-ρ n i=1 E[|ξ i | 2+ρ |F i-1 ] exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) ≤ c 1 ε ρ n exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) . Lemma 3.1 implies that for 1 ≤ x = o(γ -1 n ), E exp xM n - 1 8 x 2 [M ] n - 3 8 x 2 M n -c x 2+ρ n i=1 E[|ξ i | 2+ρ |F i-1 ] ≤ E exp xM n-1 - 1 8 x 2 [M ] n-1 - 3 8 x 2 M n-1 -c x 2+ρ n-1 i=1 E[|ξ i | 2+ρ |F i-1 ] ≤ 1.
By conditions (A1), (A2) and the last inequality, we obtain for 1

≤ x = o(γ -1 n ), E[e xMn-[M ]nx 2 /8 ] ≤ exp 3 8 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) .
Thus, from (4.25), we deduce that for 1 ≤

x = o(γ -1 n ), P (M n , [M ] n ) ∈ E 1 ≤ c 2 (ε ρ n + x 2+2ρ ε 2ρ n + δ 2 n ) exp - 1 2 x 2 - 1 4 x + c (x 2+ρ ε ρ n + x 2 δ 2 n ) ≤ c 3 (ε ρ n + δ 2 n ) exp - 1 2 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) . (4.27)
Similarly, we have For the last term P((M n , [M ] n ) ∈ E 3 ), we obtain the following estimation

P (M n , [M ] n ) ∈ E 2 ≤ (δ n /2 + 1/(2x)) -2 e -inf (u,v)∈E 2 (xu-2(vx) 2 ) E[([M ] n -1) 2 e xMn-2[M ]nx 2 ] ≤ c 4 (ε ρ n + δ 2 n ) exp - 1 2 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) . ( 4
P (M n , [M ] n ) ∈ E 3 = P M n ≥ x [M ] n , [M ] n > 16 ≤ 2 3 x -2/3 exp - 3 4 x 2 , (4.29)
where the last line follows by Lemma 3.8. Moreover, by Lemma 3.9, it holds that for ρ ∈ (0, 1],

P (M n , [M ] n ) ∈ E 3 ≤ P |[M ] n -M n | ≥ 1 ≤ c ε ρ n .
By the last inequality and (4.29), we get for 1 ≤

x = o(γ -1 n ), P (M n , [M ] n ) ∈ E 3 ≤ min c ε ρ n , 2 3 x -2/3 e -3x 2 /4 ≤ c ε ρ(2-ρ)/4 n x ρ(2+ρ)/4 exp - 1 2 x 2 . (4.30)
Thus, combining the inequalities (4.24), (4.27), (4.28) and (4.30) together, we deduce that for 1

≤ x = o(γ -1 n ), P W n ≥ x, |[M ] n -M n | > δ n + 1/(2x) ≤ c ε ρ(2-ρ)/4 n x ρ(2+ρ)/4 + δ 2 n exp - 1 2 x 2 + c (x 2+ρ ε ρ n + x 2 δ 2 n ) . ( 4
≤ x = o(γ -1 n ), P(W n ≥ x) 1 -Φ (x) ≤ 1 + c 1 (1 + x) ε ρ(2-ρ)/4 n x ρ(2+ρ)/4 + δ 2 n ×            exp c 1,ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x)(x ρ γ ρ n + γ ρ n + δ n ) if ρ ∈ (0, 1) exp c 2 x 3 ε n + x 2 δ 2 n + (1 + x) (xγ n + γ n | ln γ n | + δ n ) if ρ = 1 ≤            exp c ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x) x ρ γ ρ n + γ ρ n + δ n + ε ρ(2-ρ)/4 n x ρ(2+ρ)/4 if ρ ∈ (0, 1) exp c x 3 ε n + x 2 δ 2 n + (1 + x) xγ n + γ n | ln γ n | + δ n + ε ρ(2-ρ)/4 n x ρ(2+ρ)/4 if ρ = 1,
which gives the desired inequalities.

For the case 0 ≤ x < 1, the assertion of Theorem 2.3 follows by a similar argument, but with 1/(2x) replaced by 1/2 in (4.17) and (xε n ) ρ/2 replaced by ε ρ/2 n in (4.18), and accordingly in the subsequent statements. This completes the proof of Theorem 2.3.

Proof of Theorem 2.1

Using Proposition 3.2, by an argument similar to the proofs of Theorems 2.2 and 2.3, we obtain the following result. If ρ ∈ (0, 1), then for 0

≤ x = o(min{ε -1 n , κ -1 n }), P(W n ≥ x) 1 -Φ (x) = exp θc ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x) x ρ/2 ε ρ/2 n + ε ρ/(3+ρ) n + δ n + ε ρ(2-ρ)/4 n 1 + x ρ(2+ρ)/4 .
Notice that the following three inequalities hold:

x 1+ρ/2 ε ρ/2 n ≤ x 2+ρ ε ρ n , x ≥ ε -ρ/(2+ρ) n , x ρ/2 ε ρ/2 n ≤ ε ρ/(3+ρ) n , 0 ≤ x ≤ ε -ρ/(2+ρ) n , ε ρ(2-ρ)/4 n ≤ ε ρ/(3+ρ) n , ρ ∈ (0, 1].
Therefore, for ρ ∈ (0, 1) and 0 ≤ x = o(min{ε -1 n , κ -1 n }),

P(W n ≥ x) 1 -Φ (x) = exp θc ρ x 2+ρ ε ρ n + x 2 δ 2 n + (1 + x) ε ρ/(3+ρ) n + δ n ,
which gives the desired equality for ρ ∈ (0, 1). Assume that condition (A2) holds for ρ ≥ 1. When ρ ∈ [1, 2], by Markov's inequality and (4.22), we have for x ≥ 1,

E x 1 (xεn) 1/2 <|[M ]n-M n| ≤ 1 (xε n ) (2+ρ)/4 E x |[M ] n -M (x) n | (2+ρ)/2
≤ 1 x (2+ρ)/4 ε (3ρ-2)/4 n ≤ ε (3ρ-2)/4 n .

(4.32)

When ρ > 2, Lemma 3.7 implies that condition (A2) also holds for ρ = 2, with the term ε n in condition (A2) replaced by 2ε n . Then (4.32) with ρ = 2 shows that

E x 1 (xεn) 1/2 <|[M ]n-M n | ≤ 2ε n .
Thus, for ρ ≥ 1, it holds that

E x 1 (xεn) 1/2 <|[M ]n-M n | ≤ max ε (3ρ-2)/4 n , 2ε n ≤ 2ε ρ/(3+ρ) n .
Notice that Lemma 3.7 also implies that condition (A2) holds for ρ = 1. Therefore, by (4.20), (4.23) can be improved to

I 2 (x) ≤ exp 1 - 1 2 x 2 + c 3 x 2+ρ ε ρ n + x 2 δ 2 n E x 1 (xεn) 1/2 <|[M ]n-M n | ≤ c ε ρ/(3+ρ) n exp - 1 2 x 2 + c 3 x 3 ε n + x 2 δ 2 n .
imsart-bj ver. 2014/10/16 file: Cramer_type_large_deviations_for_martingales-180621.tex date: July 26, 2019

Notice also that for ρ ≥ 1,

P (M n , [M ] n ) ∈ E 3 ≤ min c 1 ε ρ n , 2 3 x -2/3 e -3x 2 /4 ≤ c 2 ε ρ/(3+ρ) n exp - 1 2 x 2 .
By an argument similar to the proof for case ρ ∈ (0, 1) but with the term (xε n ) ρ/2 in (4.18) replaced by (xε n ) 1/2 , we have for 0 ≤ x = o(min{ε -1 n , κ -1 n }),

P(W n ≥ x) 1 -Φ (x) = exp θc 3 x 3 ε n + x 2 δ 2 n + (1 + x) x ρ/2 ε ρ/2 n + ε ρ/(3+ρ) n + δ n = exp θc 4 x 3 ε n + x 2 δ 2 n + (1 + x) ε ρ/(3+ρ) n + δ n ,
which gives the desired equality for ρ ≥ 1.

Proof of Corollary 2.2

To prove Corollary 2.2, we need the following two sides bound on the tail probabilities of the standard normal random variable:

1 √ 2π(1 + x) e -x 2 /2 ≤ 1 -Φ(x) ≤ 1 √ π(1 + x)
e -x 2 /2 , x ≥ 0. (4.1) See p. 17 in [START_REF] Itô | Difussion Processes and Their Sample Paths[END_REF] or [START_REF] Talagrand | The missing factor in Hoeffding's inequalities[END_REF] x 2 2 -ε 1 .

Because ε 1 can be arbitrarily small, we obtain (4.3). This completes the proof of Corollary 2.2.

  bj ver. 2014/10/16 file: Cramer_type_large_deviations_for_martingales-180621.tex date: July 26, 2019

  .28)

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  For any given Borel set B ⊂ R, let x 0 = inf x∈B |x|. Then, it is obvious that x 0 ≥ inf x∈B |x|. Therefore, by Theorem 2.1 and Remark 2.1,W n ≥ a n x 0 ≤ 2 1 -Φ (a n x 0 ) × exp c ρ (a n x 0 ) 2+ρ ε ρ n + (a n x 0 ) 2 δ 2 n + (a n x 0 ) (ε ρ/(3+ρ)We may assume that B o = ∅. For any ε 1 > 0, there exists anx 0 ∈ B o , such that For x 0 ∈ B o , there exists small ε 2 ∈ (0, x 0 ), such that (x 0 -ε 2 , x 0 + ε 2 ] ⊂ B.Then it is obvious that x 0 ≥ inf x∈B x. Without loss of generality, we may assume that x 0 > 0. By Theorem 2.1, we deduce thatW n ∈ (a n (x 0 -ε 2 ), a n (x 0 + ε 2 )] ≥ P W n > a n (x 0 -ε 2 ) -P W n > a n (x 0 + ε 2 ) . -ε 2 ) 2 .

	Next, we prove that
		lim inf n→∞	1 a 2 n	ln P	W n a n	∈ B ≥ -inf x∈B o	x 2 2	.	(4.3)
						0 <	x 2 0 2	≤ inf x∈B o	x 2 2	+ ε 1 .	(4.4)
	P ≥ P Using Theorem 2.1 and (4.1), it follows that W n a n ∈ B
	lim inf n→∞ (x 0 Letting ε 2 → 0, we get 1 a 2 n ln P W n a n ∈ B ≥ -1 2
	lim inf n→∞	1 a 2 n	ln P		W n a n	∈ B ≥ -	x 2 0 2	. First, we prove that ≥ -inf x∈B o
		lim sup n→∞	1 a 2 n	ln P	W n a n	∈ B ≤ -inf x∈B	x 2 2	.	(4.2)
	P ≤ P Using (4.1), we deduce that W n a n ∈ B
	lim sup n→∞	1 a 2 n	ln P		W n a n	∈ B ≤ -	x 2 0 2	≤ -inf x∈B	x 2 2	,
	which gives (4.2).

n + δ n ) .

imsart-bj ver. 2014/10/16 file: Cramer_type_large_deviations_for_martingales-180621.tex date: July 26, 2019 X.

Fan et al.

imsart-bj ver. 2014/10/16 file: Cramer_type_large_deviations_for_martingales-180621.tex date: July 26, 2019

Acknowledgements

Fan and Liu have been partially supported by the National Natural Science Foundation of China (Grant nos. 11601375, 11626250, 11571052, 11401590 and 11731012), and by Hunan Natural Science Foundation (China, grant no. 2017JJ2271). Grama and Liu have benefitted from the support of the French government "Investissements d Avenir" program ANR-11-LABX-0020-01. Shao has been partially supported by Hong Kong RGC GRF 14302515.

Supplementary Material

Supplement to "Self-normalized Cramér type moderate deviations for martingales" (doi: COMPLETED BY THE TYPESETTER; .pdf). The supplement gives the detailed proofs of Propositions 3.1 and 3.2.

imsart-bj ver. 2014/10/16 file: Cramer_type_large_deviations_for_martingales-180621.tex date: July 26, 2019


Yuejiao Wang
email: wangyuejiaohujing@163.com

Zaiming Liu

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

Yingqiu Li

Asymptotic properties of a branching random walk with a random environment in time

Keywords: Branching random walk, random environment, quenched moments, weighted moments

published or not. The documents may come

Introduction and main results

The model of branching random walk has been studied by many authors, see e.g. [START_REF] Biggins | Martingale convergence in the branching random walk[END_REF][START_REF] Biggins | Uniform convergence of martingale in the branching random walk[END_REF][START_REF] Hu | Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees[END_REF][START_REF] Shi | Branching random walks[END_REF][START_REF] Gao | Exact convergence rates in central limit theorems for a branching random walk with a random environment in time[END_REF][START_REF] Chen | On large deviation probabilities for empirical distribution of branching random walks: Schröder case and Böttcher case[END_REF] and the references therein. A branching random walk with a random environment in time is an important extension in which the offspring distribution of a particle of generation n, and the distribution of the displacements of their children, depend on the environment ξ = (ξ) n indexed by the time n, cf. e.g. [START_REF] Biggins | Measure change in multitype branching[END_REF][START_REF] Liu | Branching random walks in random environment[END_REF][START_REF] Gao | Exact convergence rates in central limit theorems for a branching random walk with a random environment in time[END_REF][START_REF] Gao | Central limit theorems for a branching random walk with a random environment in time[END_REF].

The model of branching random walk with a random environment in time can be described as follows. As usual, let

N = {0, 1, 2, • • • }, N + = {1, 2, • • • }, R = {-∞, ∞} and U = ∞ n=0 (N + ) n
be the set of all finite sequences, where (N) +0 = {∅} contains the null sequence ∅. Let ξ = (ξ n ) n≥0 be a sequence of independent and identically distribution random variables taking values in some space Θ; each realization of ξ n corresponds probability distribution

η n = η(ξ n ) on N × R × R × • • • .
Here ξ n represents the random environment at time n. When the environment sequence ξ is given, the branching random walk starts from an initial particle ∅ of generation 0 located at the origin S ∅ = 0 ∈ R. It gives birth to N ∅ = N children of the first generation whose number and displacements (relative to their parent ∅) L ∅i = L i constitute a point process (N ;

L 1 , L 2 , • • • ) with distribution η 0 = η(ξ 0 ) on N × R × R × • • • . 1
In general, when the environment ξ is given, each particle u = u 1 • • • u n of n-th generation with position S u gives birth to N u children with displacements L u,i , so that the position of the i-th child is

S ui = S u + L ui , where (N u ; L u1 , L u2 , • • • ) has distribution η n = η(ξ n ) on N × R × R × • • • .
Conditioned on the environment ξ, all particles behave independently, which means that the family of the random vectors (N u ; L u1 , L u2 , • • • ), indexed by all finite sequences u, are conditionally independent.

Let T be the Galton-Watson tree with defining elements {N u : u ∈ U }:

(i) ∅ ∈ T; (ii) if u ∈ T, then ui ∈ T if only if 1 ≤ i ≤ N u ; (iii) if ui ∈ T, then u ∈ T.
The family {S u , u ∈ T} is called a branching random walk with a random environment in time. In the following it will be termed simply as a branching random walk in a random environment.

Let

Z n = |u|=n δ Su
be the counting measure of particles of generation n, so that for a subset A of R, Z n (A) is the number of particles of generation n located in A :

Z n (A) = |u|=n δ Su (A) = |u|=n I A (S u ),
where δ Su denotes the Dirac measure at S u and I A the indicator function of A; by convention the summation is over all particles u of generation n. The total probability space on which all the random variables ξ n and L ui , |u| = n ≥ 0 are defined will be denoted by (Ω, F, P); the conditional probability given the environment ξ will be denoted by P ξ . Therefore, by definition, for each realization of the environment sequence ξ, the random variables L ui (|u| = n ≥ 0, i ≥ 1) are independent of each other under P ξ . The probability P is usually called annealed law, while P ξ is called quenched law. The expectation with respect to P and P ξ will be denoted respectively by E and E ξ .

Fix t ∈ R. Write

m n (t) = E ξ Nu i=1 e -tL ui for |u| = n,
and assume that m n (t) < ∞. We are interested in the Laplace transform of the counting measure Z n and the associated martingale:

Ẑn (t) = e -tx Z n (dx) = |u|=n e -tSu , n ≥ 0, W 0 (t) = 1, W n (t) = Ẑn (t) E ξ Ẑn (t) = 1 n-1 i=1 m i (t) |u|=n e -tSu for n ≥ 1.
It is well known that (W n (t)) n≥0 is a nonnegative martingale under P ξ with respect to the filtration

F 0 = σ(ξ), F n = σ (ξ, N u , L u1 , L u2 , • • • , |u| < n) for n ≥ 1,
so that the limit W (t) = lim n→∞ W n (t) exists almost surely (a.s.) with E ξ W (t) ≤ 1 by Fatou's Lemma. For simplicity, write

W n = W n (t) for n ≥ 0 and W = W (t).
The necessary and sufficient conditions for the non-degeneration of W have been known: see Proposition 2.1 below applied to A ui = e -tL ui . The existence of annealed moments and weighted moments of Y have been studied in [START_REF] Liang | Asymptotic properties of the Mandelbrot's Martingale and the branching random walks[END_REF], see Theorems 2.1 and 2.2 cited below.

Here we are interested in the quenched moments and weighted moments of W , when W is non-degenerate.

We first consider the existence of the quenched moments of W .

Theorem 1.1 Let α > 1, and assume that P(W = 0) < 1.

(i) If E log + E ξ W α 1 < ∞ and E log m 0 (αt) m α 0 (t) ∈ (-∞, 0), then E ξ W α < ∞ a.s.. (ii) If E log m 0 (αt) m α 0 (t) ∈ (-∞, 0) and E ξ W α < ∞ a.s., then E log + E ξ W α 1 < ∞. (iii) If E(log -E ξ W 1 ) 2+ε < ∞ for some ε > 0, E log + E ξ W α 1 < ∞, E log m 0 (αt) m α 0 (t) 2 < ∞ and E ξ W α < ∞ a.s., then E log m 0 (αt)
m α 0 (t) < 0. Part (i) gives sufficient conditions for the existence of quenched moments E ξ W α , while parts (ii) and (iii) show that these conditions are also necessary under some additional assumptions.

Recall that a positive and measurable function l is defined on [0, ∞) is called slowly varying at ∞ if lim x→∞ l(λx) l(x) = 1 for all λ > 0. (Throughout this paper, the term "positive" is used in the wide sense.) By the representation theorem (see [START_REF] Bingham | Regular Variation[END_REF],Theorem 1.3.1), any function l slowly varying at ∞ is of the form

l(x) = c(x) exp x a 0 ε(t) t dt , x > a 0 , (1.1)
where a 0 > 0, c(•) and ε(•) are measurable with c(x) → c for some constant c ∈ (0, ∞) and ε(x) → 0 as x → ∞. Moreover, it is known that any slowly varying function l posses a smoothed version l 1 in the sense that l(x) ∼ l 1 (x) as x → ∞, with l 1 of the form

l 1 (x) = c exp x a 0 ε 1 (t) t dt , x > a 0 , (1.2)
with ε 1 infinitely differentiable on (a 0 , ∞) and lim x→∞ ε 1 (x) = 0 (see [START_REF] Bingham | Regular Variation[END_REF], Theorem 1.3.3). The value of a 0 and those of l(x) on [0, a 0 ] will not be important. For convenience, we often take a 0 = 1. Notice also that the function c(•) in the representation of l(•) has no influence on the finiteness of moments of W of the form E ξ W α l(W ), so that we can suppose without loss of generality that c(x) = 1. Moreover, by choosing a smoothed version if necessary, we can suppose that the function ε in the representation form (1.1) is infinitely differentiable.

We next give a description of the quenched weighted moments of W . Write W * = sup n≥1 W n .

Theorem 1.2 Assume that P(W = 0) < 1 and E log m 0 (αt) m α 0 (t) ∈ (-∞, 0). Let l(x) be a function slowly varying at ∞ and φ(x) = x α l(x) with α > 1. Then the following assertions are equivalent:

(i) E log + E ξ φ(W 1 ) < ∞; (ii) E ξ φ(W ) < ∞ a.s.; (iii) E ξ φ(W * ) < ∞ a.s..
The rest of the paper is organized as follows. In Section 2 we describe the model of Mandelbrot's martingale in a random environment, and state for this martingale a variante of Theorems 1.1 and 1.2: see Theorems 2.3 and 2.4, which imply Theorems 1.1 and 1.2. In Section 3, we introduce some lemmas in order to prove our main results. In Sections 4 and 5, we give respectively the proof of Theorems 2.3 and 2.4.

Mandelbrot's martingale in a random environment

The theorems stated in Section 1 will be proved for a slightly different but essentially equivalent model, i.e., for Mandelbrot's martingale in a random environment. This model is described as follows. Let ξ = (ξ n ) n≥0 be a sequence of independent and identically distribution random variables taking values in some space Θ. Suppose that when the environment ξ is given, ), defined on some probability space (Γ, P ξ ); each

{(N u , A u1 , A u2 , • • • ) : u ∈ U } is a sequence of independent random variables with values in N × R N + + , where R + = [0, ∞
(N u , A u1 , A u2 , • • • ) has distribution η(ξ n ) for |u| = n. For simplicity, we write (N, A 1 , A 2 , • • • ) for (N ∅ , A ∅1 , A ∅2 , • • • ).
Set

m n = E ξ Nu i=1
A ui , where |u| = n, n ≥ 0.

X ∅ = 1, X u = A u 1 A u 1 u 2 • • • A u 1 •••un n-1 i=0 m i , if u = u 1 • • • u n ∈ U, for n ≥ 1. Y 0 = 1 and Y n = |u|=n X u , for n ≥ 1.
Then, under P ξ , the sequence (Y n ) n≥0 forms a nonnegative martingale with respect to the filtration We can image that each node of the tree T is marked with the vector (N u , A u1 , A u2 , • • • ), A ui being associated with the edge (u, ui) linking u and ui for u ∈ T and 1 ≤ i ≤ N u ; the values of A ui for i > N u are of no influence for our results and will be taken as 0 for convenience. So the model is also called weighted branching process in a random environment. See for example [START_REF] Kuhlbusch | On weighted branching process in random environment[END_REF].

G 0 = σ(ξ) and G n = σ(ξ, N u , A u1 , A u2 , • • • , |u| < n) for n ≥ 1. It follows that (Y n , G n ) is
Remark 2.1 If A u ≡ 1 for all u, then (Y n ) becomes the natural martingale of the branching process in a random environment, studied by many authors: see for example [START_REF] Athreya | On branching processes in random environments. I: Extinction probabilities[END_REF][START_REF] Athreya | On branching processes in random environments. II: Limit theorems[END_REF][START_REF] Athreya | Branching processes[END_REF].

We are interested in the asymptotic properties of Mandelbrot's martingale in a random environment. For the existence of moments of Y in the Mandelbrot's martingale, Liu [START_REF] Liu | On generalized multiplicascades[END_REF] proved that EY α < ∞ if and only if EY α 1 < ∞ and E N i=1 A α i < 1. In this paper, we extend this result to the Mandelbrot's martingale in a random environment for the quenched law, for which we will show that the existence condition is quite different to the annealed case.

Another interest is the existence of the weighted moments of Y of the form E ξ Y α l(Y ), where α > 1 and l is a positive function slowly varying at ∞. For a Galton-Watson process, Bingham and Doney [START_REF] Bingham | Asymptotic properties of supercritical branching processes. I: The Galton-Watson process[END_REF] showed that when α > 1 is not an integer,

EY α l(Y ) < ∞ if and only if EY α 1 l(Y 1 ) < ∞.
Alsmeyer and Rösler [START_REF] Alsmeyer | On the existence of φ-moments of the limit of a normalized supercritical Galton-Watson process[END_REF] proved that the same result remains true for all non-dyadic integer α > 1 (not of the form 2 k for some integer k ≥ 0). Liang and Liu [START_REF] Liang | Weighted moments for the limit of a normalized supercritical Galton-Watson process[END_REF] proved that the result holds true for all α > 1. For the Mandelbrot's martingale, Alsmeyer and Kuhlbusch [START_REF] Alsmeyer | Double martingale structure and existence of φ-moments for weighted branching processes[END_REF] showed that when

α ∈ {2 n : n ≥ 1}, EY α l(Y ) < ∞ if and only if EY α 1 l(Y 1 ) < ∞.
Liang and Liu [START_REF] Liang | Weighted moments for Mandelbrot's martingales[END_REF] proved that the same result remains true for all α > 1 and l is a positive function slowly varying at ∞. In [START_REF] Liang | Asymptotic properties of the Mandelbrot's Martingale and the branching random walks[END_REF], this result was further extended to the Mandelbrot's martingale in a random environment for the annealed weighted moments of Y of the form EY α l(Y ). In this paper, we consider the quenched weighted moments of Y for the Mandelbrot's martingale with a random environment; we will see that the existence condition is quite different to the annealed case.

For any x ≥ 0, write

ρ ξn (x) = E ξ Nu i=1 A x ui m x n for |u| = n, n ≥ 0. ρ(x) = Eρ ξ 0 (x) = E N i=1 A x i m x 0 and ρ (x) = E N i=1 A x i m x 0 log A i m 0 if the expression is well defined (with value in [-∞, ∞]
). We mention that here ρ (x) is a notation, which coincides with the derivative of ρ(•) at x under natural regularity conditions.

For the non-degeneration of Y , a necessary and sufficient condition was shown by Biggins and Kyprianou(2004,Theorem 7.1) and Kuhlbusch (2004,Theorem 2.5).

Proposition 2.1 (Non-degeneration [START_REF] Biggins | Measure change in multitype branching[END_REF], [START_REF] Kuhlbusch | On weighted branching process in random environment[END_REF]) Assume that ρ (1) is well-defined with value in [-∞, ∞). Then the following assertions are equivalent:

(i) ρ (1) < 0 and EY 1 log + Y 1 < ∞. (ii) EY = 1. (iii) P(Y = 0) < 1.
Necessary and sufficient conditions for the existence of annealed moments and weighted moments of Y have been known. Let us recall them in the following two theorems.

Theorem 2.1 ( [START_REF] Liang | Asymptotic properties of the Mandelbrot's Martingale and the branching random walks[END_REF]) Assume P(Y = 0) < 1. For α > 1, the following assertion are equivalent:

(i) EY α 1 < ∞ and ρ(α) < 1; (ii) EY * < ∞; (iii) 0 < EY α < ∞. Theorem 2.2 ([22]) Assume that P(Y = 0) < 1 and ρ(α) < 1 for α > 1. Let l : [0, ∞) → [0, ∞)
be a function slowly varying at ∞. Then the following assertions are equivalent:

(i) EY α 1 l(Y 1 ) < ∞; (ii) EY * α l(Y * ) < ∞; (iii) 0 < EY α l(Y ) < ∞,
Here we are interested in the quenched moments and weighted moments of Y , when Y is non-degenerate. We first consider the quenched moments.

Theorem 2.3 Let α > 1, and assume that P(Y = 0) < 1. (i) If E log + E ξ Y α 1 < ∞ and E log ρ ξ 0 (α) ∈ (-∞, 0), then E ξ Y α < ∞ a.s.. (ii) If E log ρ ξ 0 (α) ∈ (-∞, 0) and E ξ Y α < ∞ a.s., then E log + E ξ Y α 1 < ∞. (iii) If E(log -E ξ Y 1 ) 2+ε < ∞ for some ε > 0, E log + E ξ Y α 1 < ∞, E log 2 ρ ξ 0 (α) < ∞ and E ξ Y α < ∞ a.s., then E log ρ ξ 0 (α) < 0.
Part (i) gives sufficient conditions for the existence of quenched moments E ξ Y α , while parts (ii) and (iii) show that these conditions are also necessary under some additional assumptions.

Notice that since (Y n ) is a nonnegative martingale under P ξ , the existence of quenched moments E ξ Y α is equivalent to the convergence in L α under P ξ .

We next consider the existence of the quenched weighted moments of Y . Theorem 2.4 Assume that P(Y = 0) < 1 and E log ρ ξ 0 (α) ∈ (-∞, 0). Let l(•) be a function slowly varying at ∞ and φ(x) = x α l(x) with α > 1. Then the following assertions are equivalent:

(i) E log + E ξ φ(Y 1 ) < ∞; (ii) E ξ φ(Y ) < ∞ a.s.; (iii) E ξ φ(Y * ) < ∞ a.s..
Clearly, Theorems 1.1 and 1.2 come from Theorems 2.3 and 2.4 with A ui = e -tL ui .

Preliminary lemmas

For the proof of our main results Theorems 1.1 and 1.2, we will use the following Lemmas. Lemma 3.1 Let (α n , β n ) n≥0 be a stationary and ergodic sequence of non-negative random variables.

(1) If E log α 0 < 0 and

E log + β 0 < ∞, then ∞ n=0 α 0 • • • α n-1 β n < ∞ a.s.. (3.1)
(2) Conversely, we have:

(a) if E log α 0 ∈ (-∞, 0) and (α n , β n ) n≥0 is i.i.d., then (3.1) implies that E log + β 0 < ∞; (b) if E| log β 0 | < ∞, then (3.1) implies that E log α 0 ≤ 0; (c) if E| log β 0 | < ∞ and E(log -β 0 ) 2+ε < ∞ for some ε > 0, then (3.1) implies that E log α 0 < 0, provided that E(log α 0 ) 2 < ∞ and that (α n ) n is i.i.d.
For Part [START_REF] Athreya | On branching processes in random environments. I: Extinction probabilities[END_REF] and the first two conclusions of Part (2), see ( [START_REF] Huang | Convergence in L p and its exponential rate for a branching process in a random environment[END_REF], Lemma 3.1). See also [START_REF] Grintsevichyus | On the continuity of the distribution of a sum of dependent variables connected with independent walks on lines[END_REF] and [START_REF] Li | Age-dependent Branching processes in random environments[END_REF] for a discussion about the convergence of the series (3.1). Below we give a proof of the last conclusion of Part (2).

Proof of Lemma 3.1. As mentioned above, we only need to prove (c) of Part [START_REF] Athreya | On branching processes in random environments. II: Limit theorems[END_REF]. By (b) of Part ( 2), we know that E log α 0 ≤ 0. Assume that E log α 0 = 0. Since E(log -β 0 ) 2+ε < ∞, we have for any constant c > 0,

∞ n=3 P(β n < exp (-c n log log n)) = ∞ n=3 P(log -β n > c n log log n) ≤ ∞ n=3 E(log -β 0 ) 2+ε (c √ n log log n) 2+ε < ∞. (3.2)
So by Borel-Cantelli's lemma, we have a.s.

β n ≥ exp (-c n log log n) for all n large enough. (3.3)
On the other hand, since σ 2 := E(log α 0 ) 2 < ∞, by the law of iterated logarithm, we have lim sup

n→∞ n-1 i=0 log α i σ 2 n log log n = 1 a.s.. (3.4)
Therefore, choosing 0 < c < σ 2 , we see that a.s., for infinitely many n,

α 0 α 1 • • • α n-1 β n ≥ exp(( σ 2 -c) n log log n) → ∞, (3.5)
which is a contradiction with (3.1). This shows that E log α 0 < 0.

For the proof of our main results, we will use the Burkholder-Davis-Gundy(BDG) inequalities that we are going to state in the following lemma. For a martingale sequence {(f n , G n ) : n ≥ 1} defined on some probability space (Ω, G, P), set f 0 = 0, G 0 = {∅, Ω},

d n = f n -f n-1 for n ≥ 1, f * = sup n≥1 |f n | and d * = sup n≥1 |d n |. Lemma 3.2 ([14],Theorem 2) Let Φ : [0, ∞) → [0, ∞)
be an increasing and continuous function with Φ(0) = 0 and Φ(2λ) ≤ cΦ(λ) for some c ∈ (0, ∞) and all λ > 0.

(i) For every β ∈ (1, 2], there exists a constant B = B c,β ∈ (0, ∞) depending only on c and β such that for any martingale {(f n , G n ) : n ≥ 1}, we have

EΦ(f * ) ≤ BEΦ(s(β)) + BEΦ(d * ) with s(β) = ∞ n=1 E |d n | β |G n-1 1/β (3.6)
and If, additionally, for some β ∈ (0, 2] the function

EΦ(f * ) ≤ BEΦ(s(β)) + B ∞ n=1 EΦ(|d n |). (3.7) (ii) If Φ is convex on [0, ∞),
Φ 1/β (x) = Φ(x 1/β ) is subadditive on [0, ∞), then EΦ(f * ) ≤ B ∞ n=1 EΦ(|d n |).
4 Proof of Theorem 2.3

Without loss of generality, we assume that m n = 1 a.s.. Otherwise we can consider Ãui := A ui /m n instead of A ui , for |u| = n, n ≥ 0. For x > 0 and n ≥ 0, write

Y n (x) := |u|=n X x u and P n (x) := E ξ |u|=n X x u = ρ ξ 0 (x) • • • ρ ξ n-1 (x). Obviously, Y n = Y n (1) and E ξ Y 1 (x) = E ξ N i=1 A x i = ρ ξ 0 (x). Let Y n+1 -Y n = |u|=n X u (Y 1 (u) -1),
where

Y 1 (u) = |v|=1 A uv , P ξ (Y 1 (u) ∈ •) = P T n ξ (Y 1 ∈ •) for |u| = n. Let D n = sup k≥1 A uk , |u| = n and D * n = sup |u|=n X u ≤ D 0 • • • D n-1 . Proof of Theorem 2.3 Notice that since (Y n ) is a martingale under P ξ , E ξ Y α < ∞ a.s. is equivalent to sup n E ξ Y α n < ∞ a.s.. Obviously, the condition sup n E ξ Y α n < ∞ a.s. is equivalent to sup n E ξ |Y n -1| α < ∞ a.s.. We first prove part (i). Suppose that E log + E ξ Y α 1 < ∞. For α ∈ (1, 2]
, the result has been proved by [START_REF] Huang | Limit theorems and the convergence rate of some branching processes and branching random walk[END_REF]. So we assume α > 2. Using BDG-inequality, we have

sup n E ξ |Y n -1| α ≤ C E ξ ∞ n=0 (Y n+1 -Y n ) 2 α 2 . (4.1) Since E ξ ∞ n=0 (Y n+1 -Y n ) 2 α 2 2 α ≤ ∞ n=0 (E ξ |Y n+1 -Y n | α ) 2 α , we have E ξ ∞ n=0 (Y n+1 - Y n ) 2 α 2 ≤ ∞ n=0 (E ξ |Y n+1 -Y n | α ) 2 α α 2 .
Therefore, together with (4.1), we have

sup n E ξ |Y n -1| α ≤ ∞ n=0 (E ξ |Y n+1 -Y n | α ) 2 α α 2 . (4.2)
So in order to prove sup n E ξ |Y n -1| < ∞ a.s., we only need to prove that ∞ n=1 (E ξ |Y n+1 -Y n | α ) 2/α < ∞ a.s.. By BDG-inequality and the convexity of x α 2 , we get

E ξ |Y n+1 -Y n | α ≤ C E ξ |u|=n X 2 u (Y 1 (u) -1) 2 α 2 = C E ξ |u|=n X 2 u Y n (2) Y n (2)(Y 1 (u) -1) 2 α 2 ≤ C E ξ |u|=n X 2 u Y n (2) Y n (2)(Y 1 (u) -1) 2 α 2 = C E ξ Y α 2 n (2)E T n ξ |Y 1 -1| α . (4.3) Since Y n (2) = |u|=n X 2 u ≤ D * n Y n (1), we have E ξ Y α 2 n (2) ≤ E ξ D * α 2 n Y α 2 n ≤ (E ξ Y α n ) 1 2 (E ξ D * α n ) 1 2 ≤ (E ξ Y α n ) 1 2 (E ξ D α 0 • • • E ξ D α n-1 ) 1 2 . (4.4)
By (4.2), (4.3) and (4.4), we have

sup n E ξ Y α n ≤ sup n E ξ (|Y n -1| + 1) α ≤ sup n 2 α (E ξ |Y n -1| α + 1) ≤ 2 α ∞ n=0 E ξ Y α 2 n (2)E T n ξ |Y 1 -1| α 2 α α 2 + 2 α ≤ 2 α sup n (E ξ Y α n ) 1 2 ∞ n=0 E ξ D α 0 • • • E ξ D α n-1 1 α E T n ξ |Y 1 -1| α 2 α α 2 + 2 α . Therefore, sup n (E ξ Y α n ) 1 2 ≤ 2 α ∞ n=0 E ξ D α 0 • • • E ξ D α n-1 1 α E T n ξ |Y 1 -1| α 2 α α 2 + 2 α / inf n (E ξ Y α n ) 1 2 . (4.5) Since 1 α E log E ξ D α 0 ≤ 1 α E log ρ ξ 0 (α) < 0 and 2 α E log E ξ Y α 1 < ∞, by Lemma 3.1, we see that ∞ n=0 E ξ D α 0 • • • E ξ D α n-1 1 α E T n ξ |Y 1 -1| α 2 α α 2 < ∞ a.s.. (4.6)
Notice that when Y > 0, then Y n > 0 for all n ≥ n 0 = n 0 (ω) large enough. Since Z k (R) = 0 implies Z n (R) = 0 for all n > k, it follows that inf n Y n > 0 a.s. on {Y > 0}, so that inf n>0 Y α n > 0 a.s. on {Y > 0}. Since P(Y > 0) > 0 implies that P ξ (Y > 0) > 0 a.s. (P ξ (Y > 0) satisfies the 0-1 law), it follows that

inf n E ξ Y α n > 0, (4.7) so that 2 α inf n (E ξ Y α n ) 1 2 < ∞ a.s.. (4.8)
From (4.5), (4.6) and (4.8), we obtain

sup n E ξ Y α n < ∞ a.s.,
so that E ξ Y α < ∞ a.s.. We next prove part (ii). Assume that E ξ Y α < ∞. We only need to prove that E log + E ξ |Y 1 -1| α < ∞. We divide the proof into two cases, according to α ≥ 2 and 1 < α < 2.

We first consider the case where α ≥ 2. By BDG-inequality and the convexity of x α 2 , we obtain

sup n E ξ |Y n -1| α ≥ C E ξ ∞ n=0 Y n+1 -Y n 2 α 2 ≥ C ∞ n=0 E ξ |Y n+1 -Y n | α ≥ C ∞ n=0 E ξ |u|=n X 2 u (Y 1 (u) -1) 2 α 2 ≥ C ∞ n=0 E ξ |u|=n X α u |Y 1 (u) -1| α = C ∞ n=0 P n (α)E T n ξ |Y 1 -1| α .
Therefore, since sup n E ξ |Y n -1| α < ∞ a.s. and E log ρ ξ 0 (α) < 0, by Lemma 3.1, we have

E log + E ξ |Y 1 -1| α < ∞, which implies E log + E ξ Y α 1 < ∞.
We next consider the case where 1 < α < 2. Let η = n P n (α). By BDG-inequality and the concavity of x α 2 , we obtain

sup n E ξ |Y n -1| α ≥ C E ξ ∞ n=0 (Y n+1 -Y n ) 2 α 2 = C E ξ ∞ n=0 P n (α) η η P n (α) (Y n+1 -Y n ) 2 α 2 ≥ C E ξ ∞ n=0 P n (α) η η α 2 P α 2 n (α) |Y n+1 -Y n | α = C η α 2 -1 ∞ n=0 P 1-α 2 n (α)E ξ |Y n+1 -Y n | α . (4.9)
For the same reason, we have

E ξ |Y n+1 -Y n | α ≥ C E ξ |u|=n X 2 u |Y 1 (u) -1| 2 α 2 = C E ξ |u|=n X 2 u Y n (2) Y n (2)|Y 1 (u) -1| 2 α 2 ≥ C E ξ |u|=n X 2 u Y n (2) Y n (2)|Y 1 (u) -1| 2 α 2 = C E ξ Y α 2 n (2)E T n ξ |Y 1 -1| α ≥ C inf n E ξ Y α 2 n (2)E T n ξ |Y 1 -1| α . (4.10)
By (4.9) and (4.10), we have

sup n E ξ |Y n -1| α ≥ C η α 2 -1 inf n E ξ Y α 2 n (2) ∞ n=0 P 1-α 2 n (α)E T n ξ |Y 1 -1| α . (4.11) By (4.7), we have inf n E ξ Y α/2 n (2) > 0 a.s.. Since sup n E ξ |Y n -1| α < ∞ and E log ρ ξ 0 (α) < 0, by Lemma 3.1, we obtain E log + E ξ |Y 1 -1| α < ∞, which implies E log + E ξ Y α 1 < ∞. Finally, part (iii) follows directly from Lemma 3.1, part (2)(c).
5 Proof of Theorem 2.4

For n ≥ 0, write M n = sup |u|=n X β-1 u , M = ∞ n=1 M n . Lemma 5.1 Let φ : [0, ∞) → [0, ∞)
be a convex and increasing function with φ(0) = 0 and φ(2x) ≤ cφ(x) for some constant c ∈ (0, ∞) and all x > 0.

Let β ∈ (1, 2]. If the function x → φ(x 1/β ) is convex, then E ξ φ(|Y * -1|) ≤ C ∞ n=1 E ξ M n-1 M φ M 1/β Y 1/β n-1 E T n-1 ξ |Y 1 | β 1/β +E ξ |u|=n-1 X β u Y n-1 (β) φ Y 1/β n-1 (β)|Y 1 (u) -1| , (5.1)
where C > 0 is a constant depending only on c and β.

Proof By (3.6), we have

E ξ φ(Y * -1) ≤ C E ξ φ ∞ n=1 E ξ |Y n -Y n-1 | β |F n-1 1 β + ∞ n=1 E ξ φ(|Y n -Y n-1 |) , (5.2)
where C > 0 is a constant only depending on c and β. By BDG-inequality, the concavity of x β 2 and the definition of M n , we obtain

E ξ (|Y n -Y n-1 | β |F n-1 ) ≤ C E ξ |u|=n-1 X 2 u (Y 1 (u) -1) 2 β 2 F n-1 ≤ C E ξ |u|=n-1 X β u |Y 1 (u) -1| β F n-1 = C |u|=n-1 X β u E T n-1 ξ |Y 1 -1| β ≤ C M n-1 Y n-1 E T n-1 ξ |Y 1 -1| β . (5.3)
where

Y 1 (u) = |v|=1 A uv , P ξ (Y 1 (u) ∈ •) = P T n-1 ξ (Y 1 ∈ •) for |u| = n -1. By (5.
3), using the fact that ∞ n=1 M n M -1 = 1 and the convexity of φ x 1 β , we have

E ξ φ ∞ n=1 E ξ (|Y n -Y n-1 | β |F n-1 ) 1 β ≤ E ξ φ ∞ n=1 CM n-1 Y n-1 E T n-1 ξ |Y 1 -1| β 1 β = E ξ φ ∞ n=1 C M n-1 M M Y n-1 E T n-1 ξ |Y 1 -1| β 1 β ≤ C E ξ ∞ n=1 M n-1 M φ M 1 β Y 1 β n-1 E T n-1 ξ |Y 1 -1| β 1 β .
(5.4)

By BDG-inequality and the convexity of φ(x

β ), we obtain

E ξ φ(|Y n -Y n-1 |) ≤ C E ξ φ |u|=n-1 X β u |Y 1 (u) -1| β 1 β = C E ξ φ |u|=n-1 X β u Y n-1 (β) Y n-1 (β)|Y 1 (u) -1| β 1 β ≤ C E ξ |u|=n-1 X β u Y n-1 (β) φ Y 1 β n-1 (β)|Y 1 (u) -1| . (5.5)
By (5.2), (5.4) and (5.5), we obtain (5.1).

For the proof of Theorem 1.2, we will use the following lemma.

Lemma 5.2 Let X be a non negative random variable, l be a function slowly varying at ∞ and φ(x) = x α l(x) with α > 1. The following assertions are equivalent:

(i) E log + E ξ φ(X) < ∞ ; (ii) E log + E ξ φ(|X -c|) < ∞, where c > 0 is a constant. Proof of Theorem 2.4 Let β ∈ (1, 2] and β < α. Write φ(x) = x α l(x)
, we can assume that the functions φ and x → φ x 1/β are convex on [0, ∞), and l(x) > 0 for all x ≥ 0 (see [START_REF] Liang | Weighted moments of the limit of a branching process in a random environment[END_REF]). Moreover, by choosing a smoothed version if necessary, we can suppose that l is differentiable.

The equivalence between (ii) and (iii) is obvious following Theorem 2.1 in [START_REF] Wang | Quenched weighted moments of a supercritical branching process in a random environment[END_REF]. The rest of the proof is composed of the following two steps.

Step 1: prove that (i) implies (iii). Suppose that E log + E ξ φ(Y 1 ) < ∞. By Lemma 5.1, we have

E ξ φ(Y * -1) ≤ C ∞ n=1 (I 1 (n) + I 2 (n)), (5.6)
where

I 1 (n) = E ξ M n-1 M φ M 1 β Y 1 β n-1 E T n-1 ξ |Y 1 -1| β 1 β , I 2 (n) = E ξ |u|=n-1 X β u Y n-1 (β) φ Y 1 β n-1 (β)|Y 1 (u) -1| .
Hence, in order to prove E ξ φ(|Y * -1|) < ∞ a.s., we only need to prove that ∞ n=1 I 1 (n) < ∞ a.s. and ∞ n=1 I 2 (n) < ∞ a.s.. We first prove that ∞ n=1 I 1 (n) < ∞ a.s.. Since l is bounded away from 0 and ∞ on any compact subset of [0, ∞), by Potter's theorem (see [START_REF] Bingham | Regular Variation[END_REF]), for ε > 0, there exists

C = C(l, ε) > 1 such that l(x) ≤ C max x ε , x -ε for all x > 0. Therefore ∞ n=1 I 1 (n) = ∞ n=1 E ξ M n-1 M α β -1 Y α β n-1 E T n-1 ξ |Y 1 -1| β α β l M 1 β Y 1 β n-1 (E T n-1 ξ |Y 1 -1| β ) 1 β ≤ C ∞ n=1 (I + 1 (n) + I - 1 (n)), (5.7)
where

I + 1 (n) = E ξ M n-1 M α+ε β -1 Y α+ε β n-1 E T n-1 ξ |Y 1 -1| β α+ε β , I - 1 (n) = E ξ M n-1 M α-ε β -1 Y α-ε β n-1 E T n-1 ξ |Y 1 -1| β α-ε β .
Choose ε 1 > 0 and ε 2 > 0 small enough such that α -ε 1 > β and (α+ε 2 )(α-ε 2 )

α-

(β+1)ε 2 β-1 ∈ (1, α + γ),
where γ is defined below (see (5.9)). Let ε = min(ε 1 , ε 2 ). Then α -ε > β and (α+ε)(α-ε)

α-

(β+1)ε β-1 ∈
(1, α + γ). Using Hölder's inequality twice and Jensen's inequality, we obtain

I + 1 (n) ≤ E ξ M n-1 M α+ε β -1 p 1 1 p 1 E ξ Y α+ε β p 2 n-1 1 p 2 E T n-1 ξ |Y 1 -1| β α+ε β ≤ E ξ M p 1 q 1 n-1 1 p 1 q 1 E ξ M α+ε β -1 p 1 q 2 1 p 1 q 2 E ξ Y α+ε β p 2 n-1 1 p 2 E T n-1 ξ |Y 1 -1| β α+ε β ≤ E ξ Y α+ε β p 2 n-1 1 p 2 E ξ M p 1 q 1 n-1 1 p 1 q 1 E ξ M α+ε β -1 p 1 q 2 1 p 1 q 2 E T n-1 ξ |Y 1 -1| α-ε α+ε α-ε , (5.8) where p 2 = β(α-ε) α+ε , q 1 = α+ε β and 1 p 1 + 1 p 2 = 1 q 1 + 1 q 2 = 1. As E log ρ ξ 0 (x) is convex for E log ρ ξ 0 (1) = 0 and E log ρ ξ 0 (α) < 0, there exists some γ ∈ (0, 1) such that E log ρ ξ 0 (x) < 0 for x ∈ (1, α + γ];
(5.9) in particular,

E log ρ ξ 0 (α -ε) < 0 for 0 < ε < α -1.
(5.10) By Potter's theorem, for ε > 0, there exists

C = C(l, ε) > 0 such that l(x) ≥ Cx -ε for all x ≥ 1. Therefore E ξ |Y 1 -1| α-ε ≤ 1 + 1 C E ξ φ(|Y 1 -1|) implies that E log + E ξ |Y 1 -1| α-ε ≤ E log + E ξ 1 + 1 C E ξ φ(|Y 1 -1|) ≤ 1 + log + 1 C + E log + E ξ |Y 1 -1| α l(Y 1 -1) < ∞. ( 5
E ξ Y α-ε n-1 < ∞ a.s.. (5.12) Since p 1 q 1 (β -1) = (α+ε)(α-ε) α- (β+1)ε β-1 ∈ (1, α + γ).
we have E log ρ ξ 0 (p 1 q 1 (β -1)) < 0. So by the triangular inequality for the norm p 1 q 1 in L p 1 q 1 , we have

E ξ M p 1 q 1 n 1 p 1 q 1 ≤ E ξ |u|=n X β-1 u p 1 q 1 1 p 1 q 1 ≤ E ξ |u|=n X (β-1)p 1 q 1 u 1 p 1 q 1 = P n-1 (β -1)p 1 q 1 1 p 1 q 1 (5.13)
and

E ξ M α+ε β -1 p 1 q 2 1 p 1 q 2 = (E ξ M p 1 q 1 ) 1 p 1 q 2 ≤ ∞ n=1 (E ξ M p 1 q 1 n-1 ) 1 p 1 q 1 α+ε β -1 ≤ ∞ n=1 P n-1 ((β -1)p 1 q 1 ) 1 p 1 q 1 α+ε β -1 < ∞ a.s.. (5.14)
By (5.8) and (5.13), we have

∞ n=1 I + 1 (n) ≤ sup n≥1 E ξ Y α-ε n-1 E ξ M α+ε β -1 p 1 q 2 1 p 1 q 2 ∞ n=1 E ξ M p 1 q 1 n-1 1 p 1 q 1 E T n-1 ξ |Y 1 -1| α-ε α+ε α-ε ≤ sup n≥1 E ξ Y α-ε n-1 E ξ M ( α+ε β -1)p 1 q 2 1 p 1 q 2 ∞ n=1 P n-1 ((β -1)p 1 q 1 ) 1 p 1 q 1 E T n-1 ξ |Y 1 -1| α-ε α+ε α-ε .
Therefore, since E log ρ ξ 0 (p 1 q 1 (β -1)) < 0 and E log + E ξ Y α-ε 1 < ∞, by Lemmas 3.1 and 5.2, together with (5.12) and (5.14), we have

∞ n=1 I + 1 (n) < ∞ a.s.. (5.15)
By an argument similar to that used above for the case of I + 1 (n), choosing ε > 0 small enough such that α -ε > β, we have

I - 1 (n) ≤ (E ξ M p 3 n-1 M p 3 ( α-ε β -1) ) 1 p 3 (E ξ Y p 4 α-ε β n-1
)

1 p 4 (E T n-1 ξ |Y 1 -1| β ) α-ε β ≤ E ξ M p 3 q 3 n-1 1 p 3 q 3 E ξ M ( α-ε β -1)p 3 q 4 1 p 3 q 4 E ξ Y α-ε n-1 1 p 4 E T n-1 ξ |Y 1 -1| α-ε = E ξ M p 3 q 3 n-1 1 p 3 q 3 E ξ M p 3 q 3 1 p 3 q 3 ( α-ε β -1) E ξ Y α-ε n-1 1 p 4 E T n-1 ξ |Y 1 -1| α-ε
≤ C(P n-1 ((β -1)p 3 q 3 ))

1 p 3 q 3 (E ξ Y α-ε n-1 ) 1 p 4 E T n-1 ξ |Y 1 -1| α-ε = C(P n-1 (α -ε)) 1 p 3 q 3 (E ξ Y α-ε n-1 ) 1 p 4 E T n-1 ξ |Y 1 -1| α-ε ,
where

q 3 = (α -ε)/β > 1, p 3 = β β-1 > 1, 1 p 3 + 1 p 4 = 1 q 3 + 1 q 4 = 1. Therefore ∞ n=1 I - 1 (n) ≤ C ∞ n=1 (P n-1 (α -ε)) 1 p 3 q 3 (E ξ Y α-ε n-1 ) 1 p 4 E T n-1 ξ |Y 1 -1| α-ε ≤ C sup n (E ξ Y α-ε n-1 ) 1 p 4 ∞ n=1 (P n-1 (α -ε)) 1 p 3 q 3 E T n-1 ξ |Y 1 -1| α-ε . (5.16)
Hence, since E log ρ ξ 0 (α -ε) < 0 and E log + E ξ Y α-ε 1 < ∞, by Lemmas 3.1 and 5.2 together with (5.12), we have

∞ n=1 I - 1 (n) < ∞ a.s.. (5.17)
By (5.7), (5.15) and (5.17), we obtain

∞ n=1 I 1 (n) < ∞ a.s.. (5.18)
We next prove

∞ n=1 I 2 (n) < ∞ a.s.
. By Potter's theorem, for ε > 0, there exists C = C(l, ε) > 0 such that l(xy) ≤ Cl(x) max{y ε , y -ε }. Hence, we obtain

I 2 (n) = C E ξ |u|=n-1 X β u Y n-1 (β) Y α β n-1 (β)|Y 1 (u) -1| α l Y 1 β n-1 (β)|Y 1 (u) -1| ≤ C(I + 2 (n) + I - 2 (n)). (5.19)
where

I + 2 (n) = E ξ Y α+ε β n-1 (β)E T n-1 ξ φ(|Y 1 -1|), I - 2 (n) = E ξ Y α-ε β n-1 (β)E T n-1 ξ φ(|Y 1 -1|).
In order to prove ∞ n=1 I 2 (n) < ∞ a.s., we only need to prove that ∞ n=1 I + 2 (n) < ∞ a.s. and

∞ n=1 I - 2 (n) < ∞ a.s.. We now consider ∞ n=1 I + 2 (n). By M n = sup |u|=n X β-1
u and Höler's inequality, we have

I + 2 (n) ≤ E ξ M n-1 Y n-1 α+ε β E T n-1 ξ φ(|Y 1 -1|) ≤ E ξ M α+ε β p n-1 1 p E ξ Y α+ε β p * n-1 1 p * E T n-1 ξ φ(|Y 1 -1|), where p * = β(α-ε) α+ε > 1, 1 p + 1 p * = 1. Therefore, we have ∞ n=1 I + 2 (n) ≤ C ∞ n=1 E ξ M α+ε β p n-1 1 p (E ξ Y α+ε β p * n-1 ) 1 p * E T n-1 ξ φ(|Y 1 -1|) ≤ C ∞ n=1 E ξ |u|=n-1 X (β-1) α+ε β p u 1 p E ξ Y α+ε β p * n-1 1 p * E T n-1 ξ φ(|Y 1 -1|) ≤ C sup n≥1 (E ξ Y α-ε n-1 ) 1 p * ∞ n=1 P n-1 (p) 1 p E T n-1 ξ φ(|Y 1 -1|).
where ε > 0 is small enough such that p := α+ε β (β -1)p ∈ (1, α + γ), then E log m 0 (p) < 0. Since E log ρ ξ 0 (p) < 0 and E log + E ξ φ(|Y 1 -1|) < ∞, by Lemma 3.1 and the above display, we have

∞ n=1 I + 2 (n) < ∞ a.s.. (5.20)
We now consider ∞ n=1 I - 2 (n). Similar to the case of I + 2 (n), we have

I - 2 (n) = E ξ Y n-1 (β) α-ε β E T n-1 ξ φ(|Y 1 -1|) ≤ E ξ (M n-1 Y n-1 ) α-ε β E T n-1 ξ φ(|Y 1 -1|) ≤ E ξ M α-ε β q n-1 1 q E ξ Y α-ε β q * n-1 1 q * E T n-1 ξ φ(|Y 1 -1|),
where q * = β, q = β β-1 and ε ∈ (0, α -1). Therefore, we have

∞ n=1 I - 2 (n) ≤ ∞ n=1 E ξ M α-ε β q n-1 1 q E ξ Y q * α-ε β n-1 1 q * E T n-1 ξ φ(|Y 1 -1|) ≤ sup n≥1 E ξ Y α-ε n-1 1 q * ∞ n=1 E ξ M α-ε β q n-1 1 q E T n-1 ξ φ(|Y 1 -1|) ≤ sup n≥1 E ξ Y α-ε n-1 1 q * ∞ n=1 P n-1 (α -ε) 1 q E T n-1 ξ φ(|Y 1 -1|).
Therefore, since E log ρ ξ 0 (α -ε) < 0 and E log + E ξ φ(|Y 1 -1|) < ∞, by Lemma 3.1, we have

∞ n=1 I - 2 (n) < ∞ a.s.. ( 5
ξ φ(|Y * -1|) < ∞ a.s. which is equivalent to E ξ φ(Y * ) < ∞ a.s..
Step 2: prove that (iii) implies (i). Assume that E ξ φ(Y * ) < ∞ a.s.. We only need to prove that E log + E ξ φ(|Y 1 -1|) < ∞ by Lemma 5.2. We divide the proof into two cases, according to α > 2 and 1 < α ≤ 2.

(a) We first consider the case α > 2. By Lemma 3.2(ii) and the convexity of φ 1/2 (x) = φ(x 1/2 ), we obtain

E ξ φ(|Y * -1|) ≥ AE ξ φ ∞ n=1 (Y n -Y n-1 ) 2 1 2 ≥ A ∞ n=1 E ξ φ ((Y n -Y n-1 ) 2 ) 1 2 = A ∞ n=1 E ξ φ(|Y n -Y n-1 |).
For the same reason, we have

E ξ φ(|Y n -Y n-1 |) ≥ AE ξ φ |u|=n-1 X 2 u (Y 1 (u) -1) 2 1 2 ≥ AE ξ |u|=n-1 φ X u |Y 1 (u) -1| .
Choose ε > 0 small enough such that α-ε > 1. By Potter's theorem, there exists C = C(l, ε) such that l(xy) ≥ C l(x) min{y ε , y -ε }, for all x > 0 and y > 0. Hence, we have

E ξ φ(|Y * -1|) ≥ A ∞ n=1 E ξ ( |u|=n-1 φ(X u |Y 1 (u) -1|)) ≥ AC ∞ n=1 E ξ ( |u|=n-1 X α u |Y 1 (u) -1| α min(X ε u , X -ε u )l(|Y 1 (u) -1|)) = AC ∞ n=1 E ξ |u|=n-1 X α u min(X -ε u , X ε u )E T n-1 ξ φ(|Y 1 -1|) = AC ∞ n=1 min P n-1 (α -ε), P n-1 (α + ε) E T n-1 ξ φ(|Y 1 -1|).
Since -∞ < E log ρ ξ 0 (α) < 0 for α > 1, by the law of large numbers, we have for δ > 0 and n > n(ξ) large enough, P n-1 (α -ε) > e (E log ρ ξ 0 (α-ε)-δ)(n-1) , P n-1 (α + ε) > e (E log ρ ξ 0 (α+ε)-δ)(n-1) , then there is a(α, ε) ∈ (-∞, 0) such that a.s. P n-1 (α -ε) ≥ e a(α,ε)(n-1) and P n-1 (α + ε) ≥ e a(α,ε)(n-1) for n > n(ξ) large enough. Therefore the preceding lower bound of Eφ(Y * -1) implies that ∞ n=1 e a(α,ε)(n-1) E T n-1 ξ φ(|Y 1 -1|) < ∞ a.s.. Hence, by Lemma 3.1 we obtain E log + E ξ φ(|Y 1 -1|) < ∞ which, by Lemma 5.2, implies E log + E ξ φ(Y 1 ) < ∞ . (b) We next consider the case where 1 < α ≤ 2. Let F = ∞ n=1 P n-1 (α). By BDGinequality and the concavity of φ 1/2 (x) = φ(x 1/2 ), we have

E ξ φ(|Y * -1|) ≥ CE ξ φ ∞ n=1 (Y n -Y n-1 ) 2 1 2 = CE ξ φ 1 2 ∞ n=1 (Y n -Y n-1 ) 2 = CE ξ φ1 2 ∞ n=1 P n-1 (α) F F P n-1 (α) (Y n -Y n-1 ) 2 ≥ CE ξ ∞ n=1 P n-1 (α) F φ 1 2 F P n-1 (α) (Y n -Y n-1 ) 2 = C ∞ n=1 P n-1 (α) F E ξ φ F 1 2 P -1 2 n-1 (α)|Y n -Y n-1 | .
By BDG-inequality and the concavity of φ 1/2 (x), we get n-1 [START_REF] Athreya | On branching processes in random environments. II: Limit theorems[END_REF] . By (4.9), we have inf n>0 Y n (2) > 0 a.s.. Hence U > 0 a.s. on {Y > 0}. Since P ξ (Y > 0) > 0, this implies that E ξ U > 0. Since ∞ < E log ρ ξ 0 (α) < 0, by the law of large numbers, there is a ∈ (∞, 0) such that a.s. P n-1 (α) ≥ e (n-1)a for n ≥ n(ξ) large enough. Therefore the preceding lower bound of E ξ φ(Y * -1) implies that ∞ n=1 e (n-1)a(1+ α+ε

E ξ φ(F 1 α P -1 2 n-1 (2)|Y n -Y n-1 |) ≥ CE ξ φ |u|=n-1 F 1 P n-1 (α) X 2 u |Y 1 (u) -1| 2 1 2 = CE ξ φ1 2 |u|=n-1 F 1 P n-1 (α) X 2 u Y n-1 (2) Y n-1 (2)|Y 1 (u) -1| 2 ≥ CE ξ |u|=n-1 X 2 u Y n-1 (2) φ F
2 ) E T n-1 ξ φ(|Y 1 -1|) < ∞ a.s.. Hence, by Lemma 3.1, we get E log + E ξ φ(|Y 1 -1|) < ∞ which, by Lemma 5.2, implies that E log + E ξ φ(Y 1 ) < ∞ . Thus we have proved that (iii) implies (i).

2 n 1 / 2 ;

 212 then there exist constants A = A c ∈ (0, ∞) and B = B c ∈ (0, ∞), depending only on c, such that for any martingale {(f n , G n ) : n ≥ 1}, we have AEΦ(S) ≤ EΦ(f * ) ≤ BEΦ(S), where S = ∞ n=1 d moreover, for any β ∈ (0, 2], EΦ(f * ) ≤ BEΦ(S(β)), where S(β) =

n- 1 ( 2 ) 2 n- 1 2 n- 1 2 inf n Y α+ε 2 n- 1 ( 2

 1221212212 |Y 1 (u) -1| . By Potter's theorem, for ε > 0, there existsA = A(l, ε) > 0 such that l(xy) ≥ Al(y) min{x ε , x -ε }. Therefore E ξ φ(|Y * -1|) (α) φ(|Y 1 (u) -1|) (α) E T n-1 ξ φ(|Y 1 -1|) E T n-1 ξ φ(|Y 1 -1|),where U = min F α+ε

Acknowledgements

The work has benefited from the support of the French government Investissements dAvenir program ANR-11-LABX-0020-01. It has been partially supported by the National Natural Science Foundation of China (Grants no. 11571052, no. 11401590, no. 11731012 and no. 11671404 ), and by Hunan Natural Science Foundation (China, grant no. 2017JJ2271).