On the rate of convergence in the central limit theorem for martingale difference sequences

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On the rate of convergence in the central limit theorem for martingale difference sequences

Lahcen Ouchti

LMRS, UMR 6085, université de Rouen, site Colbert, 76821 Mont-Saint-Aignan cedex, France Received 7 July 2003; received in revised form 5 February 2004; accepted 17 March 2004

Available online 11 September 2004

Abstract

We established the rate of convergence in the central limit theorem for stopped sums of a class of martingale difference sequences.

2004 Elsevier SAS. All rights reserved.

Résumé

On établit la vitesse de convergence dans le théorème limite central pour les sommes arrêtées issues d’une classe de suites de différences de martingale.

2004 Elsevier SAS. All rights reserved.

MSC: 60G42; 60F05

Keywords: Central limit theorem; Martingale difference sequence; Rate of convergence

1. Introduction

Let(X_i)_i∈N be a sequence of random variables defined on a probability space (Ω,F,P). We shall say that (Xk)k∈Nis a martingale difference sequence if, for anyk0

1. E{|Xk|}<+∞.

2. E{X_k+1|Fk} =0, whereFkis theσ-algebra generated byX_i,ik.

For each integern1 andxreal number, we denote

E-mail address: lahcen.ouchti@univ-rouen.fr (L. Ouchti).

0246-0203/$ – see front matter 2004 Elsevier SAS. All rights reserved.

doi:10.1016/j.anihpb.2004.03.003

S₀=0, S_n= n

i=1

X_i, φ(x)= 1

√2π x

−∞

exp

−t² 2

dt, σ_n²₋₁=E{X_n²|Fn−1},

ν(n)=inf

k∈N^∗: k

i=0

σ_i²n

, S_ν(n)² = +∞

k=1

S²_kIν(n)=k, σ_ν(n)² = +∞

k=1

σ_k²Iν(n)=k, F_n(x)=P(S_ν(n)x√

n ), S_ν(n) =S_ν(n)+

γ (n)X_ν(n)+1, H_n(x)=P(S_ν(n) x√ n ), andγ (n)is a random variable such that

ν(n)−1 i=0

σ_i²+γ (n)σ_ν(n)² =n a.s. (1)

If the random variablesX_i are independent and identically distributed withEX_i =0 andEX²_i =1, we have by the central limit theorem (CLT)

n→+∞lim sup

x∈R

P(S_nx√

n )−φ(x)=0.

By the theorem of Berry ([1], 1941) and Esseen ([5], 1942), if moreover,E|X_i³|<+∞, the rate of convergence in the limit is of ordern^−1/2. If(X_i)_i∈Nis an ergodic martingale difference sequence withEX_i²=1, by the the- orem of Billingsley ([2], 1968) and Ibragimov (([11], 1963), see also ([10], 1980)) we have the CLT. The rate of convergence can, however, be arbitrarily slow even ifX_i are bounded andα-mixing (cf. [4]). There are several results showing that with certain assumption on the conditional varianceE(X²_i |Fi−1), the rate of convergence becomes polynomial (Kato ([12], 1979), Grams ([7], 1972), Nakata ([9], 1976), Bolthausen ([3], 1982), Haeusler ([8], 1988), . . . ).

In 1963, Ibragimov [11] has shown that forXi uniformly bounded, if instead of usual sumsSn, the stopped sumsSν(n) orS_ν(n) are considered, one gets the rate of convergence of ordern^−1/4; the only assumption beside boundedness is that ^+∞_i=0σ_i²diverge to infinity a.s.

In the present paper we give a rate of convergence for a larger class of martingale difference sequences, the Ibragimov’s case will be a particular one.

2. Main result

We consider a sequence(X_i)_i∈Nof square integrable martingale differences.

Theorem 1. If the series ^+∞_i=0σ_i²diverges a.s. and if there exists a nondecreasing sequence(Y_i)_i∈N adapted to the filtration(Fi, i∈N)such that, for alli∈N^∗

E(Y_i⁴) <+∞, 1Y_i and E

|X_i|³|Fi−1

Y_i−1σ_i²₋₁ a.s.

then for all n sufficiently large

sup

x∈R

F_n(x)−φ(x) an1/2

π n^1/4

11+ 3

4n^1/4+ 2

9n^1/2+ 1 8n^3/4

, (2)

sup

x∈R

Hn(x)−φ(x) a_n^1/2 π n^1/4

11+ 9

4n^1/4+ 2

9n^1/2+ 1 8n^3/4

, (3)

wherean=(EY_ν(n)⁴ )^1/2.

If we putY_i=Ma.s. whereM >0 is a constant, one obtains the following corollaries:

Corollary 1. If the series ^+∞_i=0σ_i²diverges a.s. and there existsM >0 such that, for alli∈N^∗,E(|Xi|³|Fi−1) ME(X_i²|Fi−1)a.s. then there is a constant 0< c_M<+∞

sup

x∈R

Fn(x)−φ(x) c_M

n^1/4, (4)

sup

x∈R

Hn(x)−φ(x) c_M

n^1/4. (5)

Corollary 2. If there exists 0< αM <+∞satisfyingσ_i²₋₁αandE(|X_i|³|Fi−1)M a.s. for alli∈N^∗, then there is a constant 0< c_(α,M)<+∞such that(4)and(5)hold.

Moreover, if we suppose that(Xi)i∈Nis uniformly bounded, we obtain the result of Ibragimov [11].

Corollary 3. If the series ^+∞_i=0σ_i²diverges a.s. and|Xi|M <+∞a.s. for alli0, then(4)and(5)hold.

Example. LetA=(Ak)_k∈N be a sequence of real valued random variables such that sup_k∈NE(A⁴_k)^1/4=β <

∞and consider an arbitrary sequence of variablesζ =(ζk)_k∈N∗ with zero means, unit variances, bounded third moments and which are also independent ofA. We defineX=(Ak−1ζk)_k∈N∗ andFk theσ-algebra generated by A0, A1, . . . , Ak.

Clearly(Xk,Fk, k∈N^∗)is a martingale difference sequence, and for allk∈N^∗, E(A²_k−1ζ_k²|Fk−1)=A²_k−1 a.s.,

E

|A_k−1ζ_k|³|Fk−1

|A_k−1|sup

i∈N^∗E

|ζ_i|³

A²_k−1 a.s.

If(|Ak|)_k∈Nis nondecreasing, then using Theorem 1, one obtains sup

x∈R

F_n(x)−φ(x)cβsup_k∈N∗E(|ζ_k|³)^1/4

n^1/4 ,

wherecis a positive constant.

3. Proof of theorem

According to Esseen’s theorem (see, e.g., ([6], 1954) p. 210 and ([13], 1955) p. 285), for ally >0, sup

x∈R

Fn(x)−φ(x) 1 π

y

−y

E

exp

itS_ν(n)

√n

−exp

−t² 2

dt

|t|+ 24 π√

2π y. (6)

Below we shall prove the following inequalities

E

exp

itSν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−1 a_ne^t

2 2

|t| 3√

n+ t²

4n+an|t|³ 3n^3/2 +ant⁴

4n²

, (7)

E

exp

itS_ν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−E

exp

itS_ν(n)

√n +t² 2

a_nt² 2n exp

t² 2

, (8)

E

exp

itSν(n)

√n

−E

exp

itS_ν(n)

√n

3ant²

2n , (9)

wherea_n=(EY_ν(n)⁴ )^1/2. 3.1. Proof of the inequality (7)

We have E

exp

itS_ν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−1

= +∞

k=1

E

exp

itS_k

√n + t² 2n

k−1

p=0

σ_p²

−1

Iν(n)=k

=^+∞

k=1

k

j=1

E

exp

itSj−1

√n + t² 2n

j−1

p=0

σ_p²

e

itXj√n −e⁻

t2σ2 j−1 2n

I_ν(n)=k

.

For realx, put

e^ix=1+ix+(ix)²

2 +u(x), e^−x=1−x+β(x)x²

2. (∗)

It is easily seen that, for allx∈R u(x)|x|³

6 , u(x)x²

2 , and β

|x|1.

Observing that the random variableW_jⁿ₋₁=exp(^{it S}√^j−1

n +_2n^{t2 j}_p⁻₌¹₀σ_p²)is measurable with respect to theσ-algebra Fj−1and using the identities (∗), we obtain

E

exp

itSν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−1

= +∞

k=1

k

j=1

E

W_jⁿ₋₁E itX_j

√n −t²X²_j 2n +u

tX_j

√n

+t²σ_j²₋₁ 2n +β

t²σ_j²₋₁ 2n

t⁴σ_j⁴₋₁ 8n²

Iν(n)=k

Fj−1

.

(10) Since{ν(n)=k}is measurable with respect to theσ-algebraFk, for allj 2, we have

j−1

k=1

E{X_jI_ν(n)=k|Fj−1} =

j−1

k=1

E

(X²_j−σ_j²₋₁)I_ν(n)=k|Fj−1

=0.

On the other hand, for allj1 we have

+∞

k=1

E{X_jI_ν(n)=k|Fj−1} =^+∞

k=1

E

(X²_j−σ_j²₋₁)I_ν(n)=k|Fj−1

=0.

It follows that, for allj1

kj

E{X_jI_ν(n)=k|Fj−1} =

kj

E

(X²_j−σ_j²₋₁)I_ν(n)=k|Fj−1

=0.

So, from (10) we derive

E

exp

itSν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−1

=

+∞

k=1

k

j=1

E

W_jⁿ₋₁E

u tXj

√n

+β

t²σ_j²₋₁ 2n

t⁴σ_j⁴₋₁ 8n²

I_ν(n)=k

Fj−1

+∞

k=1

k

j=1

E

exp

t² 2n

j−1

p=0

σ_p²

E

|t|³|X_j|³

6n^3/2 +t⁴σ_j⁴₋₁ 8n²

Iν(n)=k

Fj−1

. (11)

For anyj2 and any real functionψsuch thatE(ψ(Xk)) <∞for any positivek, we have

j−1

k=1

E

exp

t² 2n

j−1

p=0

σ_p²

E

ψ(Xj)Iν(n)=k|Fj−1

=

j−1

k=1

E

exp

t² 2n

j−1

p=0

σ_p²

E

ψ(X_j)|Fj−1

I_ν(n)=k

. (12)

On the other hand, for allj1, we have

+∞

k=1

E

exp

t² 2n

j−1

p=0

σ_p²

E

ψ(X_j)I_ν(n)=k|Fj−1

=E

exp

t² 2n

j−1

p=0

σ_p²

ψ(Xj)

=^+∞

k=1

E

exp

t² 2n

j−1

p=0

σ_p²

E

ψ(X_j)|Fj−1

I_ν(n)=k

. (13)

It follows from (12) and (13) that

+∞

j=1

kj

E

exp

t² 2n

j−1

p=0

σ_p²

E

ψ(X_j)I_ν(n)=k|Fj−1

=^+∞

j=1

kj

E

exp

t² 2n

j−1

p=0

σ_p²

E

ψ(X_j)|Fj−1

I_ν(n)=k

. (14)

Applying (11) and (14) forψ(x)= |x|³we deduce that

E

exp

itSν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−1

^+∞

k=1

k

j=1

E

exp

t² 2n

j−1

p=0

σ_p²

E

|t|³|X_j|³ 6n^3/2

Fj−1

I_ν(n)=k+E

t⁴σ_j⁴₋₁

8n² I_ν(n)=k Fj−1

+∞

k=1

k

j=1

E

exp

t² 2n

j−1

p=0

σ_p²

|t|³Y_j−1σ_j²₋₁

6n^3/2 I_ν(n)=k+t⁴σ_j⁴₋₁ 8n² I_ν(n)=k

. (15)

By the Hölder inequality, for allj ∈N^∗ σ_j²₋₁=E(X²_j|Fj−1)E

|X_j|³|Fj−1

2/3

Y_j^2/3₋₁σ_j^4/3₋₁ a.s., whence

σ_j²₋₁Y_j²₋₁ a.s. (16)

From (15), (16) and using the fact thatY_kY_j−11 for alljk, we deduce that

E

exp

itS_ν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−1

|t|³ 6n^3/2+ t⁴

8n² _+∞

k=1

k

j=1

E

Y_j²₋₁σ_j²₋₁exp

t² 2n

j−1

p=0

σ_p²

I_ν(n)=k

|t|³

6n^3/2+ t⁴ 8n²

_+∞

k=1

E

Y_k² k

j=1

σ_j²₋₁exp

t² 2n

j−1

p=0

σ_p²

I_ν(n)=k

. (17)

To bound up the terms appearing in (17), we will use the following elementary lemma.

Lemma 1. Letk1, then on the event{ν(n)=k}we have k

j=1

exp

t² 2n

j−1

p=0

σ_p²

t²

2nσ_j²₋₁exp t²

2

1+Y_k²t² n

.

Proof. On the event{ν(n)=k}, we have exp

t² 2

exp

t² 2n

k−1

p=0

σ_p²

−exp t²

2nσ₀²

k−1

j=1

exp

t² 2n

j−1

p=0

σ_p²

exp t²σ_j²

2n

−1

.

Using the inequality, exp(x)−1x for allx0, one obtains exp

t² 2

k−1

j=1

exp

t² 2n

j−1

p=0

σ_p²

t² 2nσ_j². Therefore

k−1

j=1

exp

t² 2n

j−1

p=0

σ_p²

t² 2nσ_j²₋₁

k−1

j=1

exp

t² 2n

j−1

p=0

σ_p²

t²

2n(σ_j²₋₁−σ_j²)+exp t²

2

=

k−2

j=1

exp

t² 2n

j

p=0

σ_p²

−exp

t² 2n

j−1

p=0

σ_p²

t²

2nσ_j²− t² 2nexp

t² 2n

k−2

p=0

σ_p²

σ_k²₋₁

+ t² 2nexp

t² 2nσ₀²

σ₀²+exp t²

2

t² 2nY_k²

k−2

j=1

exp

t² 2n

j

p=0

σ_p²

−exp

t² 2n

j−1

p=0

σ_p²

+ t² 2nY_k²exp

t² 2nσ₀²

+exp

t² 2

1+ t² 2nY_k²

exp

t² 2

.

We conclude the proof of the lemma by noting thatσ_k²₋₁Y_k²and ^k_p⁻₌₀¹σ_p²na.s. 2 Finally, according to Lemma 1 and the (17) we get

E

exp

itSν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−1

a_nexp t²

2 |t|

3√ n+ t²

4n+an|t|³ 3n^3/2 +ant⁴

4n²

,

wherean=(EY_ν(n)⁴ )^1/2. 3.2. Proof of the inequality (8)

Using (1) and the inequality|1−exp(−x)|x, for allx0 we see that

E

exp

itSν(n)

√n + t² 2n

ν(n)−1 p=0

σ_p²

−E

exp

itSν(n)

√n +t² 2

= E

exp

itSν(n)

√n +t² 2

exp

−t²

2nγ (n)σ_ν(n)²

−1

E

1−exp

−t²

2nγ (n)σ_ν(n)²

exp t²

2

E

t²

2nγ (n)σ_ν(n)²

exp t²

2

(EY_ν(n)⁴ )^1/2t² 2nexp

t² 2

. Therefore (8) holds true.

From (7) and (8) we conclude that E

exp

itS_ν(n)

√n

−exp

−t² 2

an

|t| 3√

n+3t² 4n + |t|³

3n^3/2an+ t⁴ 4n²an

.

Using Esseen’s theorem, we derive sup

x∈R

Fn(x)−φ(x)a_n π

y

−y

1 3√

n+3|t| 4n + t²

3n^3/2an+|t|³ 4n²an

dt+ 24 π√

2π y.

Hence sup

x∈R

Fn(x)−φ(x)a_n π

2y 3√

n+3y² 4n + 2y³

9n^3/2an+ y⁴ 8n²an

+ 24 π√

2πy. Choosingyin such a way thaty/√

n=1/(yan), i.e.y=(n/a²_n)^1/4, we infer that sup

x∈R

F_n(x)−φ(x) a_n^1/2 π n^1/4

11+ 3

4n^1/4+ 2

9n^1/2+ 1 8n^3/4

.

The proof of the inequality (2) in theorem is complete.

3.3. Proof of the inequality (9)

Observing that the random events{γ (n)x}∩{ν(n)=k}and consequently the random variables√

γ (n)I_ν(n)=k are measurable with respect toFk, we find that

E

exp

itS_ν(n)

√n

−E

exp

itS_ν(n)

√n

=

+∞

k=0

E

exp itSk

√n

−exp itSk

√n +it√ γ (n)

√n X_ν(n)+1

I_ν(n)=k

+∞

k=0

E

exp itS_k

√n

1−exp it√

γ (n)

√n Xν(n)+1

Iν(n)=k

=^+∞

k=0

E

exp itSk

√n

−it√ γ (n)

√n X_k+1+ t²

2nγ (n)X²_k+1−u t√

γ (n)

√n X_k+1

I_ν(n)=k

=^+∞

k=0

E

exp itS_k

√n

E

−it√ γ (n)

√n X_k+1+ t²

2nγ (n)X²_k+1−u t√

γ (n)X_k+1

√n Fk

I_ν(n)=k

=^+∞

k=0

E

exp itS_k

√n t²

2nγ (n)X²_k+1−u t

√n

γ (n)Xk+1

I_ν(n)=k

^+∞

k=0

E

I_ν(n)=k3t²

2nγ (n)X²_k+1

3t² 2n

+∞

k=0

E

I_ν(n)=kE{X²_k+1|Fk}

3t²

2nE(Y_ν(n)⁴ )^1/2.

The proof of the inequality (9) is complete.

3.4. Proof of the inequality (3)

According to Esseen’s theorem wherey=(n/a²_n)^1/4and the inequality (9), one obtains

sup

x∈R

H_n(x)−φ(x) 1 π

y

−y

E

exp

itS_ν(n)

√n

−exp

−t² 2

dt

|t|+ 24 π√

2π y a_n^1/2

π n^1/4

11+ 3

4n^1/4+ 2

9n^1/2+ 1 8n^3/4

+ 1

π y

−y

3|t|

2nE(Y_ν(n)⁴ )^1/2dt an1/2

π n^1/4

11+ 3

4n^1/4+ 2

9n^1/2+ 1 8n^3/4

+ 3

2π√ n an1/2

π n^1/4

11+ 9

4n^1/4+ 2

9n^1/2+ 1 8n^3/4

. The proof of theorem is complete. 2

Proofs of Corollaries 1, 2 and 3 are easy so, it is left to the reader.

Acknowledgement

The author thanks the referee for careful reading of the manuscript and for valuable suggestions which improved the presentation of this paper.

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