A Berry–Esseen bound of order p 1 n for martingales

Comptes Rendus

Mathématique

Songqi Wu, Xiaohui Ma, Hailin Sang and Xiequan Fan

A Berry–Esseen bound of orderp¹

n for martingales Volume 358, issue 6 (2020), p. 701-712.

<https://doi.org/10.5802/crmath.81>

© Académie des sciences, Paris and the authors, 2020.

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Creative Commons Attribution4.0International License.

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2020, 358, n6, p. 701-712

https://doi.org/10.5802/crmath.81

Probabilities /Probabilités

A Berry–Esseen bound of order ^{p 1} _n for martingales

Une borne de Berry–Esseen d’ordre ^{p 1} _n pour les martingales

Songqi Wu^a, Xiaohui Ma^a, Hailin Sang^band Xiequan Fan^∗,a

aCenter for Applied Mathematics, Tianjin University, Tianjin, China

bDepartment of Mathematics, The University of Mississippi, University, MS 38677, USA

E-mails:[email protected], [email protected], [email protected], [email protected]

Abstract.Renz [13] has established a rate of convergence 1/p

nin the central limit theorem for martin- gales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen [2] and Grama and Haeusler [6], is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.

Résumé.Renz [13] a établi un taux de convergence 1/p

ndans le théorème de la limite centrale pour les martingales avec certaines conditions restrictives. Dans le présent article, une modification des méthodes, développées par Bolthausen [2] et Grama et Haeusler [6], est appliquée pour obtenir le même taux de convergence pour une classe de martingales plus générales. Une application aux processus linéaires est discutée.

Funding.The work has been partially supported by the National Natural Science Foundation of China (Grant no. 11601375). The research of Hailin Sang is partial supported by the Simons Foundation (Grant no. 586789).

Manuscript received 23rd January 2020, accepted 2nd June 2020.

∗Corresponding author.

1. Introduction and main result

Forn ∈N, let (ξi,Fi)_i=0,...,n be a finite sequence of martingale differences defined on some probability space (Ω,F,P), whereξ0=0 and {;,Ω}=F0⊆ · · · ⊆Fn⊆F are increasingσ-fields.

Denote

X₀=0, X_k=

k

X

i=1

ξi,k=1, . . . ,n.

ThenX=(X_k,Fk)_k=0,...,nis a martingale. Denote by〈X〉the conditional variance ofX:

〈X〉0=0, 〈X〉k=

k

X

i=1

E£ ξ²_i¯

¯Fi−1

¤,k=1, . . . ,n.

Define

D(X_n)=sup

x∈R

¯¯P(X_n≤x)−Φ(x)¯

¯,

whereΦ(x) is the distribution function of the standard normal random variable. Denote by→^P the convergence in probability asn→ ∞. According to the martingale central limit theorem, the

“conditional Lindeberg condition”

n

X

i=1

E£

ξ²_i1_{|ξi|≥²}¯

¯Fi−1

¤ P

→0, for each²>0, and the “conditional normalizing condition”〈X〉n

→P 1 together implies asymptotic normality of X_n, that is,D(X_n)→0 asn→ ∞.

The convergence rate ofD(Xn) has attracted a lot of attentions. For instance, Bolthausen [2]

proved that if|ξi| ≤²nfor a number²nand〈X〉n=1 a.s., thenD(Xn)≤c²³nnlogn, where, here and after,cis an absolute constant not depending on²nandn. El Machkouri and Ouchti [3] improved the factor²³nnlognin Bolthausen’s bound to²nlognunder the following more general condition

E£

|ξi|³¯

¯Fi−1

¤≤²nE£ ξ²_i¯

¯Fi−1

¤ a.s. for alli=1, 2, . . . ,n.

For more related results, we refer to Ouchti [12] and Mourrat [11]. Recently, Fan [4] proved that if there exist a positive constantρand a number²n, such that

E£

|ξi|^2+ρ¯

¯Fi−1

¤≤²^ρnE£ ξ²_i¯

¯Fi−1

¤ a.s. for alli=1, 2, . . . ,n, and〈X〉n=1 a.s., thenD(X_n)≤c_ρb²n, where

b²n=

(²^ρn, ifρ∈(0, 1),

²n|log²n|, ifρ≥1,

andc_ρis a constant depending only onρ. Fan [4] also showed that this Berry–Esseen bound is optimal. In particular, if²n³1/p

n, then we have²n|log²n| ³(logn)/p

n. Thus, we cannot obtain the classical convergence rate 1/p

nfor general martingales.

However, the convergence rate 1/p

n for martingales is possible to be attained with some additional restrictive conditions. For instance, Renz [13] proved that if there exists a constant ρ>0 such that

E[ξ²i|Fi−1]=1/n, E[ξ³i|Fi−1]=0 and E£

|ξi|^3+ρ¯

¯Fi−1¤

≤cn^−(3+ρ)/2, a.s., (1) then it holds

D(X_n)=O µ 1

pn

¶

. (2)

He also showed that this result is not true forρ=0. More martingale Berry–Esseen bounds of convergence rate 1/p

ncan be found in Bolthausen [2] and Kir’yanova and Rotar [10].

In this paper we are interested in extending (2) to a class of more general martingales. The following theorem is our main result.

Theorem 1. Assume that there exist some numbersρ∈(0,+∞),²n∈(0,¹₂]andδn∈[0,¹₂]such that for all1≤i≤n,

¯

¯〈X〉n−1¯

¯≤δ²n, (3)

E£ ξ³_i¯

¯Fi−1

¤=0 (4)

and

E£

|ξi|^3+ρ¯

¯Fi−1

¤≤²^1+ρn E£ ξ²_i¯

¯Fi−1

¤ a.s. (5)

Then

D(X_n)≤c_ρ(²n+δn),

where c_ρdepends only onρ.In addition, it holds c_ρ=O(ρ⁻¹),ρ→0.

Notice that under the conditions of Renz [13], the conditions of Theorem 1 are satisfied withδn =0 and²n ³1/p

n. Thus Theorem 1 extends Renz’s result to a class of more general martingales.

Thanks to the additional condition (4), the Berry–Esseen bound (6) improves the bound of Fan [4] by replacing²n|log²n|with²n.

Relaxing the condition (3), we have the following analogue estimation of Fan (cf. [4, (26)]).

Theorem 2. Assume that there exist some numbersρ∈(0,+∞)and²n ∈(0,¹₂]such that for all 1≤i≤n,

E£ ξ³_i¯

¯Fi−1

¤=0 and

E£

|ξi|^3+ρ¯

¯Fi−1¤

≤²¹n^+ρE£ ξ²_i¯

¯Fi−1¤ a.s.

Then, for all p≥1,

D(X_n)≤c_ρ²n+c_p µ

E£¯

¯〈X〉n−1¯

¯

p¤ +Eh

max

1≤i≤n|ξi|^2p

i¶1/(2p+1)

, (6)

where c_ρand cpdepend only onρand p, respectively.

It is easy to see that whenp→ ∞, µ

E£¯

¯〈X〉n−1¯

¯

p¤

¶1/(2p+1)

→ k〈X〉n−1k^1/2_∞ , which coincides withδnof Theorem 1.

2. Application

We first extend Theorem 1 to triangular arrays with infinity many terms in each line. Forn∈N, let (ξn,i,Fn,i)ⁿ_i=−∞be a sequence of martingale differences defined on some probability space (Ω,F,P), where the adapted filtration is {;,Ω}=F−∞⊂ · · · ⊂Fn,n−1⊂Fn,n⊂F. DenoteX_n,k= Pk

i=−∞ξn,i,k≤n. Then (X_n,k,Fn,k)ⁿ_k=−∞is a martingale. Let〈X〉n,k=Pk

i=−∞E[ξ²_n,i|Fn,i−1],k≤n.

In particular, denoteX_n:=X_n,nand〈X〉n:= 〈X〉n,n.

With some slight modification on the proof, Theorem 1 still holds in this new setting. Now we apply Theorem 1 with this new setting to the partial sum of linear processes. Let (εi)_i∈Z be a sequence of identically distributed martingale differences adapted to the filtration (Fi)_i∈Z. We consider the causal linear process in the form

Y_k=

k

X

j=−∞

a_k−jεj, (7)

where the martingale differences have finite variance and the sequence of real coefficients satisfiesP_∞

i=0a_i²< ∞. Without loss of generality, let the variance of the martingale difference to

be 1. We say the linear process has long memory ifP_∞

i=0|a_i| = ∞. In this case, we assume that a₀=1 and

a_i=`(i)i<sup>−α</sup>,i>0, with 1/2<α<1. (8) Here`(·) is a slowly varying function. On the other hand, we say the linear process has short memory ifP_∞

i=0|a_i| < ∞andP_∞

i=0a_i6=0. The third case isP_∞

i=0|a_i| < ∞andP_∞

i=0a_i=0.

The long memory linear processes covers the well-known fractional ARIMA processes (cf.

Granger and Joyeux [7]; Hosking [9]), which play an important role in financial time series modeling and application. As a special case, let 0<d<1/2 andBbe the backward shift operator withBεk=εk−1and consider

Y_k=(1−B)^−dεk= X∞ i=0

a_iεk−i, wherea_i= Γ(i+d) Γ(d)Γ(i+1).

For this example we have limn→∞a_n/n^d−1=1/Γ(d). Note that these processes have long memory becauseP_∞

j=0|aj| = ∞. The partial sumS_n=Pn

k=1Y_kof causal linear process (7) can be written asS_n=Pn

i=−∞b_n,iεi, whereb_n,i =P_n−i

j=0a_j for 0<i ≤n, andb_n,i =P_n−i

j=1−ia_j fori ≤0. The variance ofS_n isB_n²= var(S_n)=Pn

i=−∞b²_n,i. Now let X_n,k =Pk

i=−∞b_n,iεi/B_n. ThenX_n = X_n,n =S_n/B_n and〈X〉n = P_n

i=−∞b²_n,iE[ε²_i|Fi−1]/B_n². If we assume|〈X〉n−1| ≤δ²n for someδn∈[0,¹₂],E[ε³_i|Fi−1]=0 and E[|εi|^3+ρ|Fi−1]≤d_ρ^1+ρE[ε²_i|Fi−1] a.s. for alli∈Zand some constantd_ρ, then, by Theorem 1,

sup

x∈R|P(Sn/Bn≤x)−Φ(x)| ≤c_ρ(²n+δn), where²n=d_ρsup_i≤n|bn,i|/Bn.

In the case thatP_∞

i=0|a_i| < ∞, sup_i≤n|b_n,i| ≤P_∞

i=0|a_i| < ∞and it is well known thatB_n²has ordern. Hence²n has order 1/p

n in this case. In the long memory caseP_∞

i=0|a_i| = ∞, if we assume (8),B_n²has ordern^3−2α`<sup>2</sup>(n) (e.g., Wu and Min [14]) and sup<sub>i≤n</sub>|b<sub>n,i</sub>|has ordern<sup>1−α</sup>`(n) (see Beknazaryan et al. [1] for upper bound and Fortune et al. [5] for lower bound in the case d=1). Hence in this case²n also has order 1/p

n. In either case the Berry–Esseen bound has order 1/p

nifδn=O(n^−1/2). In particular, if we in addition assume that the innovations (εi)i∈Z

are independent, thenδn=0 and the Berry–Esseen bound sup_x∈R|P(S_n/B_n≤x)−Φ(x)|has order 1/p

n. Here the conditionE[ε³_i|Fi−1]=0 is needed to have the Berry–Esseen bound of order 1/p

n. We cannot have this order from the result of Fan [4].

3. Proofs of theorems 3.1. Preliminary lemmas

In the proofs of theorems, we need the following technical lemmas. The first two lemmas can be found in Fan [4, Lemmas 3.1 and 3.2].

Lemma 3. If there exists an s>3such that

E[|ξi|^s|Fi−1]≤²^s−2n E[ξ²_i|Fi−1], then, for any t∈[3,s),

E[|ξi|^t|Fi−1]≤²^tn⁻²E[ξ²_i|Fi−1].

Lemma 4. If there exists an s>3such that

E[|ξi|^s|Fi−1]≤²^s−2n E[ξ²_i|Fi−1], then

E[ξ²_i|Fi−1]≤²²n.

The next two technical lemmas are due to Bolthausen (cf. [2, Lemmas 1 and 2]).

Lemma 5. Let X and Y be random variables. Then sup

u

¯¯P¡ X≤u¢

−Φ(u)¯

¯≤c₁sup

u

¯¯P¡

X+Y ≤u¢

−Φ(u)¯

¯+c₂°

°E[Y²|X]°

°

1/2

∞, where c₁and c₂are two positive constants.

Lemma 6. Let G(x)be an integrable function onRof bounded variationkGkV, X be a random variable and a,b6=0are real numbers. Then

E

· G

µX+a b

¶¸

≤ kGkVsup

u

¯¯P¡ X≤u¢

−Φ(u)¯

¯+ kGk1|b|, wherekGk1is the L₁(R)norm of G(x).

In the proof of Theorem 2, we also need the following lemma of El Machkouri and Ouchti [3].

Lemma 7. Let X and Y be two random variables. Then, for p≥1, D(X+Y)≤2D(X)+3°

°E£

Y^2p|X¤°

°

1/(2p+1)

1 . (9)

3.2. Proof of Theorem 1

By Lemma 3, we only need to consider the case ofρ∈(0, 1]. We follow the method of Grama and Haeusler [6]. LetT=1+δ²n. We introduce a modification of the conditional variance〈X〉nas follows:

V_k= 〈X〉k1_{k<n}+T1_{k=n}. (10)

It is easy to see thatV₀=0,V_n=T, and that (V_k,Fk)_k=0,...,nis a predictable process. Set γ=²n+δn.

Letc_∗be some positive and sufficient large constant. Define the following non-increasing dis- crete time predictable process

A_k=c_∗²γ²+T−V_k, k=1, . . . ,n. (11) Obviously, we haveA₀=c²_∗γ²+T andA_n=c_∗²γ². In addition, foru,x∈R, andy>0, denote

Φu(x,y)=Φ µu−x

py

¶

. (12)

LetN =N(0, 1) be a standard normal random variable, which is independent ofX_n. Using a smoothing procedure, by Lemma 5, we deduce that

sup

u

¯¯P¡ X_n≤u¢

−Φ(u)¯

¯≤c₁sup

u

¯¯P¡

X_n+c_∗γN ≤u¢

−Φ(u)¯

¯+c₂γ

=c1sup

u

¯

¯E£ Φu¡

Xn,An¢¤

−Φ(u)¯

¯+c2γ

≤c₁sup

u

¯¯E£ Φu

¡X_n,A_n¢¤

−E£ Φu

¡X₀,A₀¢¤¯

¯ +c1sup

u

¯

¯E£ Φu¡

X0,A0¢¤

−Φ(u)¯

¯+c2γ

=c₁sup

u

¯¯E£ Φu

¡X_n,A_n¢¤

−E£ Φu

¡X₀,A₀¢¤¯

¯

+c₁sup

u

¯

¯

¯

¯Φ

µ u

q c_∗²γ²+T

¶

−Φ(u)

¯

¯

¯

¯+c₂γ. (13) It is obvious that ¯

¯

¯

¯Φ

µ u

q c²_∗γ²+T

¶

−Φ(u)

¯

¯

¯

¯≤c₃

¯

¯

¯

¯ 1 q

c_∗²γ²+T

−1

¯

¯

¯

¯≤c₄γ. (14)

Returning to (13), we get sup

u

¯¯P¡ X_n≤u¢

−Φ(u)¯

¯≤c₁sup

u

¯¯E£ Φu

¡X_n,A_n¢¤

−E£ Φu

¡X₀,A₀¢¤¯

¯+c₅γ. (15) By a simple telescoping, we know that

E£ Φu

¡X_n,A_n¢¤

−E£ Φu

¡X₀,A₀¢¤

=E

· n

X

k=1

¡Φu

¡X_k,A_k¢

−Φu

¡X_k−1,A_k−1¢¢

¸

. (16)

Taking into account the fact that

∂²

∂x²Φu(x,y)=2 ∂

∂yΦu(x,y), we get

E£ Φu

¡X_n,A_n¢¤

−E£ Φu

¡X₀,A₀¢¤

=J₁+J₂−J₃, (17) where

J1=E

· n

X

k=1

µ

Φu(X_k,A_k)−Φu(X_k−1,A_k)− ∂

∂xΦu(X_k−1,A_k)ξk

−1 2

∂²

∂x²Φu(X_k−1,A_k)ξ²_k−1 6

∂³

∂x³Φu(X_k−1,A_k)ξ³_k

¶¸

, (18)

J₂=1 2E

· _n X

k=1

∂²

∂x²Φu(X_k−1,A_k)¡

4 〈X〉k− 4V_k¢

¸

, (19)

J₃=E

· n

X

k=1

µ

Φu(X_k−1,A_k−1)−Φu(X_k−1,A_k)− ∂

∂yΦu(X_k−1,A_k)4V_k

¶¸

, (20)

where4〈X〉k= 〈X〉k− 〈X〉k−1.

Now, we need to give some estimates ofJ1,J2andJ3. To this end, we introduce some notations.

Denote byϑi some random variables satisfying 0≤ϑi≤1, which may represent different values at different places. For the rest of the paper,ϕstands for the density function of the standard normal random variable.

Control ofJ₁. For convenience’s sake, letT_k−1=(u−X_k−1)/p

A_k,k=1, 2, . . . ,n. It is easy to see that

B_k=:Φu(X_k,A_k)−Φu(X_k−1,A_k)− ∂

∂xΦu(X_k−1,A_k)ξk

−1 2

∂²

∂x²Φu(X_k−1,A_k)ξ²_k−1 6

∂³

∂x³Φu(X_k−1,A_k)ξ³_k

=Φ µ

T_k−1− ξk

pA_k

¶

−Φ(T_k−1)+Φ⁰(T_k−1) ξk

pA_k

−1

2Φ⁰⁰(T_k−1) µ ξk

pA_k

¶2

+1

6Φ⁰⁰⁰(T_k−1) µ ξk

pA_k

¶3

. To estimate the right hand side of the last equality, we distinguish two cases.

Case 1:|ξk/p

A_k| ≤2+ |T_k−1|/2. By a four-term Taylor expansion, it is obvious that if|ξk/p A_k| ≤ 1, then

¯

¯B_k¯

¯=

¯

¯

¯

¯ 1 24Φ⁽⁴⁾

µ

T_k−1−ϑ ξk

pA_k

¶¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

4¯

¯

¯

¯

≤

¯

¯

¯

¯Φ⁽⁴⁾ µ

T_k−1−ϑ ξk

pA_k

¶¯

¯

¯

¯

¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3+ρ

.

If|ξk/p

A_k| >1, by a three-term Taylor expansion, then

¯¯B_k¯

¯≤1 2

µ¯

¯

¯

¯Φ⁰⁰⁰ µ

T_k−1−ϑ ξk

pA_k

¶¯

¯

¯

¯+

¯

¯

¯

¯Φ⁰⁰⁰(T_k−1)

¯

¯

¯

¯

¶¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3

≤

¯

¯

¯

¯Φ⁰⁰⁰ µ

T_k−1−ϑ⁰ ξk

pA_k

¶¯

¯

¯

¯

¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3

≤

¯

¯

¯

¯Φ⁰⁰⁰ µ

T_k−1−ϑ⁰ ξk

pA_k

¶¯

¯

¯

¯

¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3+ρ

, where

ϑ⁰=





 ϑ, if¯

¯Φ⁰⁰⁰¡

T_k−1−ϑp^ξk

A_k

¢¯

¯≥ |Φ⁰⁰⁰(T_k−1)|, 0, if¯

¯Φ⁰⁰⁰¡

T_k−1−ϑp^ξk

A_k

¢¯

¯< |Φ⁰⁰⁰(T_k−1)|. Using the inequality max{|Φ⁰⁰⁰(t)|,|Φ⁰⁰⁰⁰(t)|}≤ϕ(t)(2+t⁴), we find that

¯¯B_k1_{|ξ

k/p

A_k|≤2+|T_k−1|/2}

¯

¯≤ϕ µ

T_k−1−ϑ1 ξk

pA_k

¶µ 2+

µ

T_k−1−ϑ1 ξk

pA_k

¶4¶¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3+ρ

≤g₁(T_k−1)

¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3+ρ

, (21)

where

g₁(z)= sup

|t−z|≤2+|z|/2ϕ(t)(2+t⁴).

Case 2:|ξk/p

A_k| >2+ |T_k−1|/2. It is obvious that, for|4x| >1+ |x|/2,

¯

¯

¯

¯Φ(x− 4x)−Φ(x)+Φ⁰(x)4x−1

2Φ⁰⁰(x)(4x)²+1

6Φ⁰⁰⁰(x)(4x)³

¯

¯

¯

¯

≤ µ¯

¯

¯

¯

Φ(x− 4x)−Φ(x)

|4x|³

¯

¯

¯

¯+ |Φ⁰(x)| + |Φ⁰⁰(x)| + |Φ⁰⁰⁰(x)|

¶

|4x|³

≤ µ

8

¯

¯

¯

¯

Φ(x− 4x)−Φ(x) (2+ |x|)³

¯

¯

¯

¯+ |Φ⁰(x)| + |Φ⁰⁰(x)| + |Φ⁰⁰⁰(x)|

¶

|4x|³

≤ µ

ce

(2+ |x|)³+ |Φ⁰(x)| + |Φ⁰⁰(x)| + |Φ⁰⁰⁰(x)|

¶

|4x|³

≤ cb

(2+ |x|)³|4x|³

≤ cb

(2+ |x|)³|4x|^3+ρ. Hence, we have

¯¯B_k1_{|ξ

k/p

A_k|>2+|T_k−1|/2}

¯

¯≤g₂(T_k−1)

¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3+ρ

, (22)

where

g₂(z)= cb (2+ |z|)³. Denote

G(z)=g₁(z)+g₂(z).

Combining (21) and (22) together, we get

|B_k| ≤G(T_k−1)

¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3+ρ

. (23)

Therefore,

¯¯J₁¯

¯=

¯

¯

¯

¯ E

· _n X

k=1

B_k

¸¯

¯

¯

¯≤E

· _n X

k=1

G(T_k−1)

¯

¯

¯

¯ ξk

pA_k

¯

¯

¯

¯

3+ρ¸

. (24)

Next, we consider conditional expectation of|ξk|^3+ρ. By condition (5), we get E£

|ξk|^3+ρ¯

¯Fk−1

¤≤²^1+ρn 4 〈X〉k, (25)

where4〈X〉k= 〈X〉k− 〈X〉k−1and we know that

4 〈X〉k= 4V_k=V_k−V_k−1, 1≤k<n,4〈X〉n≤ 4V_n, (26) then

E£

|ξk|^3+ρ¯

¯Fk−1

¤≤²¹n^+ρ4V_k. (27)

By (24) and (27), we obtain

|J1| ≤R1:=²¹n^+ρ

· _n X

k=1

G(T_k−1) A⁽³_k^+ρ)/24V_k

¸

. (28)

To estimateR₁, we introduce the time changeτtas follow: for any realt∈[0,T],

τt=min{k≤n:V_k≥t}, where min; =n. (29) Obviously, for anyt∈[0,T], the stopping timeτtis predictable. In addition, (σk)_k=1,...,n+1(with σ1 = 0) stands for the increasing sequence of moments when the increasing and stepwise functionτt,t ∈[0,T], has jumps. It is easy to see that4V_k=R

[σk,σk+1)dt, and that k =τt for t∈[σk,σk+1). SinceτT =n, we have

n

X

k=1

G(T_k−1) A^(3+ρ_k ^)/24V_k=

n

X

k=1

Z

[σk,σk+1)

G(T_τt−1) A^(3+ρ_τt ^)/2

dt= ZT

0

G(T_τt−1) A^(3+ρ_τt ^)/2

dt. (30)

Leta_t=c_∗²γ²+T−t. Because of4V_τt≤2²²n+2δ²n(cf. Lemma 4), we know that

t≤V_τt=V_τt−1+ 4V_τt≤t+2²²n+2δ²n, t∈[0,T]. (31) Assumec_∗≥2, then we have

1

2a_t≤A_τt=c²_∗γ²+T−V_τt≤a_t, t∈[0,T]. (32) Note thatG(z) is symmetric and is non-increasing inz≥0. The last bound implies that

R₁≤2^(3+ρ)/2²^1+ρn

Z T 0

1 a_t^(3+ρ)/2

E

· G

µu−X_τt−1 a^1/2_t

¶¸

dt. (33)

Note also thatG(z) is a symmetric integrable function of bounded variation. By Lemma 6, it is obvious that

E

· G

µu−X_τt−1 a^1/2_t

¶¸

≤c₆sup

z

¯¯P¡

X_τt−1≤z¢

−Φ(z)¯

¯+c₇p

a_t. (34)

Because ofc_∗≥2,V_τt−1=V_τt− 4V_τt,V_τt≥tand4V_τt≤2²²n+2δ²n, we obtain

V_n−V_τt−1=V_n−V_τt+ 4V_τt≤2²²n+2δ²n+T−t≤a_t. (35) Therefore

E£¡

X_n−X_τt−1¢2¯

¯Fτt−1

¤=E

· _n X

k=τt

E£ ξ²_k¯

¯Fk−1

¤

¯

¯

¯

¯Fτt−1

¸

=E£

〈X〉n− 〈X〉_τt−1

¯

¯F_τt−1

¤

≤E[Vn−V_τt−1|Fτt−1]

≤a_t.

Then, by Lemma 5, we deduce that for anyt∈[0,T], sup

z

¯

¯P¡

X_τt−1≤z¢

−Φ(z)¯

¯≤c8sup

z

¯

¯P¡ Xn≤z¢

−Φ(z)¯

¯+c9p

at. (36)

Combining (28), (33), (34) and (36) together, we get

|J1| ≤c10²¹n^+ρ

ZT 0

1 a⁽³_t^+ρ)/2

dtsup

z

¯

¯P¡ Xn≤z¢

−Φ(z)¯

¯+c11²¹n^+ρ

Z T 0

1 a¹_t^+ρ/2

dt. (37)

Taking some elementary computations, it follows that ZT

0

1 a^(3+ρ_t ^)/2

dt= ZT

0

1

(c²_∗γ²+T−t)^(3+ρ)/2dt≤ 2

c_∗^1+ρ(1+ρ)γ^1+ρ (38) and

ZT 0

1 a¹_t^+ρ/2

dt= Z T

0

1

(c_∗²γ²+T−t)^1+ρ/2dt≤ 2

c^ρ_∗ργ^ρ. (39) This yields

¯¯J₁¯

¯≤ c12

c¹_∗^+ρsup

z

¯¯P¡ Xn≤z¢

−Φ(z)¯

¯+c_ρ,1²n

ρ . (40)

Control ofJ₂. Since 0≤ 4V_k− 4〈X〉k≤2δ²1_{k=n}, we have

|J₂| ≤E

· 1 2An

¯

¯ϕ⁰(T_n−1)(4V_n− 4〈X〉n)¯

¯

¸ .

DenoteG(z)e =sup_|z−t|≤1|ϕ⁰(t)|, and then|ϕ⁰(z)| ≤G(ze ) for any realz. SinceAn=c²_∗γ², then we get the following estimation:

|J2| ≤ 1 c²_∗E£

G(Te n−1)¤ .

Note thatGeis non-increasing inz ≥0, and thus it has bounded variation onR. By Lemma 6, we get

|J₂| ≤c₁₃ c_∗² sup

z

¯¯P¡

X_n−1≤z¢

−Φ(z)¯

¯+c_∗,2(²n+δn). (41) Then, by Lemma 5, we deduce that

sup

z

¯¯P¡

X_n−1≤z¢

−Φ(z)¯

¯≤c₁₄sup

z

¯¯P¡ X_n≤z¢

−Φ(z)¯

¯+c₁₅²n. (42) This yields

|J₂| ≤c₁₆ c_∗² sup

z

¯

¯P¡ X_n≤z¢

−Φ(z)¯

¯+c_ρ,2(²n+δn). (43) Control ofJ₃. By a two-term Taylor expansion, it follows that

|J₃| =1 8E

· n

X

k=1

1

(A_k−ϑk4A_k)²ϕ⁰⁰⁰

µ u−X_k−1 pA_k−ϑk4A_k

¶ (4A_k)²

¸ . Note thatc_∗≥2,4A_k≤0 and, by Lemma 4,|4A_k| = 4V_k≤2²²n+2δ²n. We obtain

A_k≤A_k−ϑk4A_k≤c_∗²γ²+T−V_k+2²²n+2δ²n≤2A_k. (44) DenoteG(z)b =sup_|t−z|≤2|ϕ⁰⁰⁰(t)|. ThenG(z) is symmetric, and is non-increasing inb z ≥0. Us- ing (44), we get

|J3| ≤(2²²n+2δ²n)E

· n

X

k=1

1 A²_kGb

µT_k−1 p2

¶ 4V_k

¸

. (45)

By an argument similar to that of (40), we get

|J₃| ≤c17(2²²n+2δ²n) c²_∗γ² sup

z

¯¯P¡ X_n≤z¢

−Φ(z)¯

¯+2c18(2²²n+2δ²n) c_∗γ

≤c19

c_∗² sup

z

¯¯P¡ Xn≤z¢

−Φ(z)¯

¯+4c18(²n+δn)² c_∗γ

≤c₁₉ c_∗² sup

z

¯

¯P¡ X_n≤z¢

−Φ(z)¯

¯+c_ρ,3(²n+δn). (46)

Combining (17), (40), (43) and (46) together, we get

¯¯E£ Φu

¡X_n,A_n¢¤

−E£ Φu

¡X₀,A₀¢¤¯

¯≤ c₂₀ c_∗^1+ρ

sup

z

¯¯P¡ X_n≤z¢

−Φ(z)¯

¯+bc_ρ

ρ(²n+δn), By (15), we know that

sup

z

¯¯P¡ X_n≤z¢

−Φ(z)¯

¯≤ c₂₁ c_∗^1+ρ

sup

z

¯¯P¡ X_n≤z¢

−Φ(z)¯

¯+ce_ρ

ρ(²n+δn), from which, choosingc_∗^1+ρ=max {2c21, 2^1+ρ}, we get

sup

z

¯

¯P¡ X_n≤z¢

−Φ(z)¯

¯≤2ce_ρ(²n+δn)

ρ . (47)

3.3. Proof of Theorem 2

Following the method of Bolthausen [2], we enlarge the sequence (ξi,Fi)_1≤i≤n to¡ ξbi,Fci

¢

1≤i≤N

such that­ Xb®

N:=PN i=1E£

ξb²_i|Fci−1¤

=1 a.s., and then apply Theorem 1 to the enlarged sequence.

Consider the stopping time

τ=sup{k≤n:〈X〉k≤1}. (48)

Assume that 0≤ε≤²n. Letr=¥_{1−〈X〉τ}

ε²

¦, wherebxcdenotes the “integer part” ofx. It is easy to see thatr≤¥₁

ε²

¦. SetN =n+r+1. Let (ζi)_i≥1be a sequence of independent Rademacher random variables, which is independent of the martingale differences (ξi)1≤i≤n. Consider the random variables¡

ξbi,Fci¢

1≤i≤Ndefined as follows:

ξbi=















ξi a.s., ifi≤τ,

εζi a.s., ifτ+1≤i≤τ+r,

¡1− 〈X〉τ−rε²¢1/2

ζi a.s., ifi=τ+r+1,

0 a.s., ifτ+r+1≤i≤N,

andFci=σ¡

ξb1,ξb2, . . . ,ξbi

¢. Clearly,¡

ξbi,Fci

¢

1≤i≤Nstill forms a martingale difference sequence with respect to the enlarged filtration. ThenXb_k=Pk

i=1ξbi,k=0, . . . ,N, withXb₀=0, is also a martingale. Moreover, it holds that

­ Xb®

N=1,E£ bξ³_i¯

¯ cFi−1

¤=0 and E£¯

¯ bξi

¯

¯^3+ρ

¯

¯ cFi−1

¤≤²^1+ρn E£ ξb²_i¯

¯ cFi−1

¤, a.s.

By Theorem 1, we have

D¡ Xb_N¢

≤c_ρ²n

ρ . (49)

Using Lemma 7, we obtain that D(X_n)≤2D¡

Xb_N¢ +3°

°E£¯

¯X_n−Xb_N¯

¯

2p¯

¯ bX_N¤°

°^1/(2p+1)

1 ≤2c_ρ²n

ρ +3¡ E£¯

¯ bX_N−X_n¯

¯

2p¤¢_1/(2p+1) . (50)