Self-normalized Cramér type moderate deviations for martingales

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martingales

Xiequan Fan, Ion Grama, Quansheng Liu, Qi-Man Shao

To cite this version:

Xiequan Fan, Ion Grama, Quansheng Liu, Qi-Man Shao. Self-normalized Cramér type moderate

deviations for martingales. Bernoulli, Bernoulli Society for Mathematical Statistics and Probability,

2019, 25 (4A), pp.2793-2823. �10.3150/18-BEJ1071�. �hal-02487825�

arXiv:arXiv:0000.0000

Self-normalized Cram´er type moderate deviations for martingales

XIEQUAN FAN^1,* ION GRAMA^2,** QUANSHENG LIU^2,† and QI-MAN SHAO^3,‡

1Center for Applied Mathematics, Tianjin University, Tianjin 300072, China.

E-mail:^*[email protected]

2Universit´e de Bretagne-Sud, LMBA, UMR CNRS 6205, Campus de Tohannic, 56017 Vannes, France. E-mail:^**[email protected]; ^†[email protected]

3Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.

E-mail:^‡[email protected]

Let (Xi,Fi)i≥1be a sequence of martingale differences. SetSn=Pn

i=1Xiand [S]n=Pn i=1X_i². We prove a Cram´er type moderate deviation expansion forP(Sn/p

[S]n≥x) asn→+∞.Our results partly extend the earlier work of [Jing, Shao and Wang,2003] for independent random variables.

Keywords:Martingales, self-normalized sequences, Cram´er’s moderate deviations.

1. 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_i²<∞for alli≥1. Set

Sn=

n

X

i=1

Xi, B_n² =

n

X

i=1

EX_i², V_n²=

n

X

i=1

X_i².

It is well-known that under the Lindeberg condition the central limit theorem (CLT) holds

sup

x∈R

P(Sn/Bn≤x)−Φ(x)

→0 asn→ ∞,

where Φ(x) denotes the standard normal distribution function. Cram´er’s moderate devi- ation expansion stated below gives an estimation of the relative error ofP(Sn/Bn≥x) to 1−Φ(x).If (X_i)_i≥1 are identically distributed with Ee^t0

√|X1|<∞for some t₀ >0 (cf. [Linnik,1961]), then for 0≤x=o(n^1/6) asn→ ∞,

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) Expansion is available for 0≤x=o(n^1/2) if the moment generating function exists. We refer to Chapter VIII of [Petrov,1975] for further details on the subject.

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 self- normalized partial sumsSn/Vn originates from Student’st-statistic. Student’st-statistic Tn is defined by

Tn=√

n Xn/bσ, where

Xn =Sn

n and σb²=

n

X

i=1

(Xi−Xn)² n−1 . It is known that for allx≥0,

P

Tn ≥x

=P

Sn/Vn≥x n n+x²−1

^1/2 ,

see [Chung, 1946]. So, if we get an asymptotic bound on the tail probabilities for self- normalized partial sums, then we have an asymptotic bound on the tail probabilities for Tn. [Gin´e, G¨otze and Mason, 1997] gave a necessary and sufficient condition for the asymptotic normality. [Slavova,1985] and [Bentkus, Bloznelis and G¨otze, 1996] (see also [Bentkus and G¨otze, 1996]) obtained the Berry-Esseen bounds for self-normalized partial sums. See also [Novak,2011] and [Shao and Wang, 2013] for Berry-Esseen type inequalities with explicit constants. [Shao, 1997] established a self-normalized Cram´er- Chernoff large deviation without any moment assumptions and [Shao, 1999] proved a self-normalized Cram´er moderate deviation theorem under (2 +ρ)th moments: if (X_i)_i≥1 are independent and identically distributed with E|X1|^2+ρ < ∞, ρ ∈ (0,1], then for 0≤x=o(n^ρ/(4+2ρ)) asn→ ∞,

P(Sn/Vn≥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 [Jing, Shao and Wang, 2003] under finite (2 +ρ)th moments,ρ∈(0,1], showing that

P(Sn/Vn ≥x)

1−Φ (x) = expn O 1

(1 +x)^2+ρι^ρ_no

(1.3) uniformly for 0≤x=o(min{ι⁻¹_n , ς_n⁻¹}),whereO(1) is bounded by an absolute constant and

ι^ρ_n=

n

X

i=1

E|Xi|^2+ρ/B^2+ρ_n and ς_n²= max

1≤i≤nEX_i²/B_n². (1.4) For further self-normalized Cram´er type moderate deviation results for independent ran- dom variables we refer, for example, to [Hu, Shao and Wang,2009], [Liu, Shao and Wang, 2013], and [Shao and Zhou,2016]. We also refer to [de la Pe˜na, Lai and Shao,2009] and [Shao and Wang,2013] for recent developments in this area.

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 Pe˜na, 1999], [Bercu and Touati, 2008], [Chen, Wang, Xu and Miao, 2014] and [Bercu, Delyon and Rio, 2015]. The main purpose of this paper is to establish self- normalized Cram´er type moderate deviation results for martingales. Let (δn)_n≥1,(εn)_n≥1 and (κn)_n≥1 be three sequences of nonnegative numbers, such thatδn→0, εn →0 and κn →0 asn→ ∞.Let (Xi,Fi)_i≥1be a sequence of martingale differences satisfying

n

X

i=1

E[X_i²|F_i−1]−B_n²

≤δ²_nB_n²,

n

X

i=1

E[|Xi|^2+ρ|Fi−1]≤ε^ρ_nB^2+ρ_n , and

1≤i≤nmax E[X_i²|F_i−1]≤κ²_nB_n²,

where ρ ∈ (0,³₂]. Here and hereafter, the inequalities between random variables are understood in theP-almost sure sense. From Corollary2.1we 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_n⁻¹, κ⁻¹_n }) asn→ ∞.A more general Cram´er type expansion is obtained in a larger range in our Theorem 2.1, from which we de- rive a moderate deviation principle for self-normalized martingales. Moreover, when the conditionPn

i=1E[|Xi|^2+ρ|F_i−1]≤ε^ρ_nB^2+ρ_n is replaced by a slightly stronger condition E[|Xi|^2+ρ|F_i−1]≤(εnBn)^ρE[X_i²|F_i−1],

equality (1.5) holds for a larger range of 0≤x=o( min{ε^{−ρ/(4+2ρ)}n , δ_n⁻¹}) forρ∈(0,1], see Corollary2.4. 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 Section2. Section 3 provides the preliminary lemmas that are used in the proofs of the main results. In Section4, we prove the main results.

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

2. Main results

Let (Xi,Fi)i=0,...,nbe a sequence of martingale differences defined on a probability space (Ω,F,P), whereX0= 0 and{∅,Ω}=F0⊆...⊆ Fn⊆ F are increasingσ-fields. Set

S0= 0, Sk =

k

X

i=1

Xi, k= 1, ..., n. (2.1)

ThenS= (S_k,F_k)_k=0,...,n is a martingale. DenoteB_n²=Pn

i=1EX_i². Let [S] andhSibe, respectively, the square bracket and the conditional variance of the martingaleS,that is

[S]0= 0, [S]k =

k

X

i=1

X_i², k= 1, ..., n, and

hSi0= 0, hSik=

k

X

i=1

E[X_i²|F_i−1], k= 1, ..., n. (2.2) In the sequel, we use the following conditions:

(A1) There exists δn ∈[0,¹₄] such that

n

X

i=1

E[X_i²|Fi−1]−B²_n

≤δ_n²B_n²; (A2) There existρ >0 andε_n∈(0,¹₄] such that

n

X

i=1

E[|Xi|^2+ρ|Fi−1]≤ε^ρ_nB_n^2+ρ; (A3) There exists κ_n ∈(0,¹₄] such that

E[X_i²|F_i−1]≤κ²_nB_n², 1≤i≤n;

(A4) There existρ∈(0,1] andγn∈(0,¹₄] such that

E[|Xi|^2+ρ|F_i−1]≤(γnBn)^ρE[X_i²|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/ργ_n. Thus, we may assume that ε_n =O(γ_n) asn → ∞. It is also easy to see that condition (A4) implies condition (A3) withκ_n =γ_n, see Lemma 3.5.

In practice, we usually have max{δn, εn, γn, κn} →0 asn→ ∞. In the case of sums of i.i.d. random variables, conditions (A1), (A2), (A3), and (A4) are satisfied withδn= 0, εn, γn, κn=O(^√1_n).

Our first main result is the following Cram´er type moderate deviation for the self- normalized martingale

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

Theorem 2.1. Assume that conditions (A1), (A2), and (A3) are satisfied. Set ρ₁= min{ρ,1}.

Then for 0≤x=o(min{ε⁻¹_n , κ⁻¹_n }), P(Wn≥x)

1−Φ (x) = exp

θcρ

x^2+ρ1ε^ρ_n¹+x²δ²_n+ (1 +x) ε^ρ/(3+ρ)_n +δn

. (2.3)

Under condition (A2) the best Berry-Esseen bound for standardized martingales is provided by [Haeusler,1988]: assuming hSin=B²_n a.s., Haeusler proved that

sup

x

P(Sn/Bn≤x)−Φ (x) ≤cρ

Xⁿ

i=1

E|Xi/Bn|^2+ρ1/(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 c0,ρ and a sequence of mar- tingale differences satisfying P(S_n ≤ 0)−Φ (0) ≥ c_0,ρ Pn

i=1E|X_i/B_n|^2+ρ1/(3+ρ) for all large enoughn. In particular, under conditions (A2) andhSi_n =B²_n a.s., Haeusler’s result implies that

sup

x

P(Sn/Bn ≤x)−Φ (x)

≤cρε^ρ/(3+ρ)_n . (2.4)

Notice that Theorem2.1implies that, for each absolute constantc >0 there is a positive constantcρdepending onρsuch that fornlarge 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 hSin =B_n² a.s., the bound in (2.5) for self-normalized mar- tingales is of the same order as the bound in (2.4) for standardized martingales.

From Theorem2.1, 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 ∈ (0,³₂]. Then

P(Wn ≥x)

1−Φ (x) = 1 +o(1)

uniformly for0≤x=o( min{ε^−ρ/(3+ρ)n , κ⁻¹_n , δ_n⁻¹})as n→ ∞.

Theorem2.1also implies the following moderate deviation principles (MDP) for self- normalized martingales.

Corollary 2.2. Assume conditions (A1), (A2), and (A3) withmax{δn, εn, κn} →0 as n→ ∞. Let an be any sequence of real numbers satisfying an → ∞ and anεn →0 as

n→ ∞. Then for each Borel setB,

− inf

x∈B^o

x²

2 ≤ lim inf

n→∞

1 a²_n lnP

Wn

a_n ∈B

≤ lim sup

n→∞

1 a²_n lnP

Wn

an

∈B

≤ − inf

x∈B

x²

2 , (2.6)

whereB^o andB denote the interior and the closure of B, respectively.

The last corollary shows that the convergence speed of MDP depends only onεn and it has nothing to do with the convergence speeds ofκn andδn.

For i.i.d. random variables, the self-normalized MDP was established by [Shao,1997].

See also [Jing, Liang and Zhou,2012] for non-identically distributed random variables.

The other main results concern some improvements of Theorem2.1 when condition (A3) is replaced by the stronger condition (A4). Theorems 2.2 and 2.3 below give re- spectively lower and upper bounds, while Theorem 2.4 gives a Cram´er type moderate deviation expansion sharper than that in Theorem2.1.

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

[i] Ifρ∈(0,1), then for0≤x=o(γ_n⁻¹), P(Wn≥x)

1−Φ (x) ≥exp

−cρ

x^2+ρε^ρ_n+x²δ²_n+ (1 +x) (x^ργ^ρ_n+γ_n^ρ+δn)

. (2.7) [ii] If ρ= 1, then for0≤x=o(γ_n⁻¹),

P(Wn≥x) 1−Φ (x) ≥exp

−c

x³εn+x²δ_n²+ (1 +x) (xγn+γn|lnγn|+δn)

. (2.8) The term γ_n|lnγ_n| in (2.8) cannot be replaced by γ_n under the stated conditions.

Indeed, [Bolthausen,1982] (see Example 2 therein) showed that there exists a sequence of martingale differences satisfying|X_i| ≤2 andhSi_n =na.s., such that for allnlarge enough,

P(Sn≥0)−Φ (0)

≥ clogn

√n , (2.9)

where c does not depend on n. Inequality (2.9) shows that the term γ_n|lnγ_n| in (2.8) cannot be replaced byγ_n even for bounded martingale differences.

For any sequence of positive numbers (αn)_n≥1 denote

αbn(x, ρ) = α^ρ(2−ρ)/4n

1 +x^ρ(2+ρ)/4. (2.10)

Accordingly, we shall use below the notationsεb_n(x, ρ) andbγ_n(x, ρ), which mean sequences defined by (2.10) withα_n replaced byε_n andγ_n.

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

[i] Ifρ∈(0,1), then for0≤x=o(γ_n⁻¹), P(Wn≥x)

1−Φ (x) ≤exp

cρ

x^2+ρε^ρ_n+x²δ_n²+ (1 +x)

x^ργ^ρ_n+γ_n^ρ+δn+εbn(x, ρ) . [ii] If ρ= 1, then for0≤x=o(γ_n⁻¹),

P(Wn≥x) 1−Φ (x) ≤exp

c

x³εn+x²δ²_n+ (1 +x)

xγn+γn|lnγn|+δn+εbn(x,1) . Combining Theorems 2.2 and 2.3, we obtain the following Cram´er type moderate deviation expansion for self-normalized martingales under conditions (A1), (A2), and (A4), which is stronger than the expansion in Theorem2.1since the termε^ρ/(3+ρ)n therein is improved to a smaller one.

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

[i] Ifρ∈(0,1), then for0≤x=o(γ_n⁻¹), P(Wn≥x)

1−Φ (x) = exp

θcρ

x^2+ρε^ρ_n+x²δ_n²+ (1 +x)

x^ργ_n^ρ+γ^ρ_n+δn+εbn(x, ρ) . [ii] If ρ= 1, then for0≤x=o(γ_n⁻¹),

P(Wn≥x) 1−Φ (x) = exp

θc

x³εn+x²δ²_n+ (1 +x)

xγn+γn|lnγn|+δn+εbn(x,1) . Notice that condition (A4) implies condition (A2) withε_n=γ_n.Therefore, it follows from Theorem2.4that:

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

[i] Ifρ∈(0,1), then for0≤x=o(γ_n⁻¹), P(Wn≥x)

1−Φ (x) = exp

θcρ

x^2+ργ_n^ρ+x²δ²_n+ (1 +x)

δn+bγn(x, ρ) . [ii] If ρ= 1, then for0≤x=o(γ_n⁻¹),

P(Wn≥x) 1−Φ (x) = exp

θc

x³γn+x²δ²_n+ (1 +x)

δn+γn|lnγn|+bγn(x,1) . From Theorem 2.4, we also obtain the following result about the equivalence to the normal tail.

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

1−Φ (x) = 1 +o(1) (2.11)

uniformly for0≤x=o( min{ε^−ρ/(2+ρ)n , γn^−ρ/(1+ρ), δ⁻¹_n })asn→ ∞.

In the case of i.i.d. random variables, conditions (A1), (A2), and (A4) are satisfied with ε_n, γ_n =O(1/√

n) andδ_n = 0. Thus, the range 0≤x=o( min{ε^−ρ/(2+ρ)n , δ_n⁻¹, γn^−ρ/(1+ρ)}) reduces to 0 ≤ x= o(n^{−ρ/(4+2ρ)}), n→ ∞, which is the best possible result such that (2.11) holds (see [Shao, 1999]). Moreover, from Theorem2.4, we can get the estimation of the rate of convergence in (2.11); for example, whenρ= 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 withc0>0forc0n^3/22≤x=o(n^1/2) asn→ ∞, P(W_n≥x)

1−Φ (x) = expn θc x³

n^1/2 o

. (2.12)

In particular, withc₀, c₁>0forc₀n^3/22≤x≤c₁n^1/6,

P(Wn ≥x) 1−Φ (x) −1

≤c x³

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,Fk)_k=0,...,n satisfies conditions (A1), (A2), (A3), and (A4), then (−Sk,Fk)_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(W_1−Φ(x)^n≥x) is replaced by

P(Wn≤−x) Φ(−x) .

3. Preliminary lemmas

The proofs of Theorems2.1-2.4are based on a conjugate multiplicative martingale tech- nique for changing the probability measure which is similar to that of the transformation of [Esscher,1924]. Our approach is inspired by the earlier work of [Grama and Haeusler, 2000] on Cram´er moderate deviations for standardized martingales, and by that of [Shao, 1999], [Jing, Shao and Wang,2003], who developed techniques for moderate deviations of self-normalized sums of independent random variables. We extend these work by intro- ducing a new choice of the density for the change of measure and refining the approaches in [Shao,1999] and [Jing, Shao and Wang, 2003] to handle self-normalized martingales.

A key point of the proof is a new Berry-Esseen bound for martingales under the changed measure, see Proposition3.1below.

Let

ξi= Xi

B_n, i= 1, ..., n.

Then (ξi,Fi)i=0,...,n is also a sequence of martingale differences. Moreover, for simplicity of notations, set

Mk =

k

X

i=1

ξi,

[M]k =

k

X

i=1

ξ_i² and hMik =

k

X

i=1

E[ξ²_i|F_i−1], k= 1, ..., n.

Thus

Wn= Sn

p[S]n

= Mn

p[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

Z0(λ) = 1, Zk(λ) =

k

Y

i=1

e^ζi(λ)

E[e^ζi(λ)|F_i−1], k= 1, ..., n with

ζi(λ) =λξi−λ²ξ²_i/2.

Thus, for each real numberλ and each k= 1, ..., n,the random variable Zk(λ) is non- negative and EZk(λ) = 1. The last observation allows us to introduce the conjugate probability measure Pλ:=Pλ,n on (Ω,F) defined by

dPλ=Zn(λ)dP. (3.2)

Although (M_k,F_k)_k=0,...,nis a martingale under the measureP,it is no longer a martin- gale under the conjugate probability measureP_λ. To obtain a martingale under P_λ we have to center the random variablesζi(λ). Denote by Eλ the expectation with respect toPλ. BecauseZ(λ) is a uniformly integrable martingale underP, we have

E_λ[ζ] =E[ζZ_n(λ)] (3.3)

and

Eλ[ζ|Fi−1] =E[ζe^ζi(λ)|Fi−1]

E[e^ζi(λ)|F_i−1] (3.4)

for anyFi-measurable random variableζthat is integrable with respect to Fi.Set b_i(λ) =E_λ[ζ_i(λ)|Fi−1], i= 1, . . . , n,

η_i(λ) =ζ_i(λ)−b_i(λ), i= 1, . . . , n, and

Yk(λ) =

k

X

i=1

ηi(λ), k= 1, ..., n. (3.5)

ThenY(λ) = (Yk(λ),Fk)k=0,...,n is theconjugate martingale. The following semimartin- gale decomposition is well-known:

k

X

i=1

ζi(λ) =Bk(λ) +Yk(λ), k= 1, ..., n, (3.6) whereB(λ) = (B_k(λ),Fk)_k=0,...,n is thedrift process defined as

Bk(λ) =

k

X

i=1

bi(λ), k= 1, ..., n.

By the relation betweenE andE_λ onFi, we have b_i(λ) = E[ζi(λ)e^ζi(λ)|F_i−1]

E[e^ζi(λ)|Fi−1] , i= 1, ..., n. (3.7) It is easy to compute the conditional variance of the conjugate martingaleY(λ) under the measurePλ, fork= 0, ..., n,

hY(λ)i_k =

k

X

i=1

Eλ[ηi(λ)²|F_i−1]

=

k

X

i=1

E_λ[(ζ_i(λ)−b_i(λ))²|Fi−1]

=

k

X

i=1

E[ζ_i²(λ)e^ζi(λ)|F_i−1]

E[e^ζi(λ)|Fi−1] −E[ζi(λ)e^ζi(λ)|F_i−1]² E[e^ζi(λ)|Fi−1]²

. (3.8)

In the sequel, we give the upper and lower bounds for Bn(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 [Shao,1999] and [Jing, Shao and Wang,2003]

established for independent random variables. Set

eε_i,λ=λ²E[ξ²_i1_{|λξi|>1}|Fi−1] +λ³E[|ξi|³1_{{|λξi|≤1}}|Fi−1], λ≥0.

IfE[|ξ_i|^2+ρ]<∞forρ∈[0,1],then it is obvious that

εei,λ≤λ^2+ρE[|ξi|^2+ρ|F_i−1], λ≥0.

Lemma 3.1. For allλ >0andτ ∈[¹₈,2],we have

E[e^{λξi−τ λ2ξi2}|Fi−1]−1−(1

2 −τ)λ²E[ξ_i²|Fi−1]

≤ceεi,λ. Lemma 3.2. For allλ >0, we have

E[e^ζi(λ)|F_i−1]−1

≤ cεei,λ,

E[ζ_i(λ)e^ζi(λ)|Fi−1]−1

2λ²E[ξ_i²|Fi−1]

≤ cεe_i,λ,

E[ζ_i²(λ)e^ζi(λ)|F_i−1]−λ²E[ξ_i²|F_i−1]

≤ cεei,λ, E[|ζi(λ)|³e^ζi(λ)|Fi−1] ≤ cεe_i,λ,

E[ζi(λ)e^ζi(λ)|F_i−1]2

≤ cεei,λ. Lemma 3.3. Let Hi=ξ_i²−E[ξ_i²|F_i−1].Then for allλ >0,

E[Hie^ζi(λ)|F_i−1]

≤ c 1 λ²εei,λ.

Using Lemma3.2, we obtain the following upper and lower bounds forB_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 }),

Bn(λ)−1

2λ²hMin

≤c λ^2+ρε^ρ_n. (3.9)

Proof. According to the definition ofbi(λ),we have bi(λ) = E[ζi(λ)e^ζi(λ)|F_i−1]

E[e^ζi(λ)|Fi−1] . By Lemma3.2, it follows that

E[ζ_i(λ)e^ζi(λ)|Fi−1]−1

2λ²E[ξ_i²|Fi−1]

≤cεe_i,λ and

E[e^ζi(λ)|Fi−1]−1

≤cεei,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

2λ²E[ξ_i²|F_i−1] ≤cεe_i,λ and

Bn(λ)−1

2λ²hMin

≤c λ^2+ρε^ρ_n as desired.

The following lemma shows that condition (A4) implies condition (A3) withκ_n=γ_n. Lemma 3.5. Assume condition (A4). ThenE[ξ²_i|F_i−1]≤γ_n².

Proof. By Jensen’s inequality and condition (A4), it holds that E[ξ_i²|F_i−1]^(2+ρ)/2≤E[|ξi|^2+ρ|F_i−1]≤γ_n^ρE[ξ²_i|F_i−1], from which we getE[ξ²_i|F_i−1]≤γ_n².

Lemma 3.6. Assume condition (A4). Then for anyt∈[0, ρ),

E[|ξi|^2+t|Fi−1]≤γ_n^tE[ξ_i²|Fi−1]. (3.11) Proof. Letl, p, q be defined by the following equations

lp= 2, (2 +t−l)q= 2 +ρ, p⁻¹+q⁻¹= 1, l >0, andp, q≥1.

Solving the last equations, we get l= 2(ρ−t)

ρ , p= ρ

ρ−t, q=ρ t. By H¨older’s inequality and condition (A4), it is easy to see that

E[|ξi|^2+t|Fi−1] = E[|ξi|^l|ξi|^2+t−l|Fi−1]

≤ (E[|ξi|^lp|F_i−1])^1/p(E[|ξi|^(2+t−l)q|F_i−1])^1/q

≤ (E[ξ_i²|F_i−1])^1/p(E[|ξ_i|^2+ρ|F_i−1])^1/q

≤ (E[ξ_i²|Fi−1])^1/p(γ_n^ρE[ξ_i²|Fi−1])^1/q

≤ γ_n^ρ/qE[ξ²_i|F_i−1]

= γ_n^t E[ξ_i²|Fi−1].

This completes the proof of the lemma.

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

n

X

i=1

E[|ξi|^2+t|F_i−1]≤2ε^t_n. (3.12)

Proof. Recall the notations in the proof of Lemma3.6. It is easy to see that

n

X

i=1

E[|ξi|^2+t|F_i−1]≤

n

X

i=1

(E[ξ²_i|F_i−1])^1/p(E[|ξi|^2+ρ|F_i−1])^1/q.

Using H¨older’s inequality and conditions (A1) and (A2), we have

n

X

i=1

E[|ξi|^2+t|Fi−1] ≤ Xⁿ

i=1

E[ξ²_i|Fi−1]^1/pXⁿ

i=1

E[|ξi|^2+ρ|Fi−1]^1/q

≤ 2ε^t_n, which gives the desired inequality.

We will also need the following two lemmas.

Lemma 3.8. Assume condition (A1). Then for allx >0, P

Mn≥xp

[M]n, [M]n≥16

≤2

3x^−2/3expn

−3 4x²o

. Proof. By inequality (11) of [Delyon,2009], we have for allλ∈R,

Eexpn

λM_n−λ² 2 (1

3[M]_n+2

3hMin)o

≤1.

Applying the last inequality to the exponential inequality of [de la Pe˜na and Pang,2009]

withp=q= 2,we deduce that for allx >0,

P |Mn|

q3

2(¹₃[M]_n+²₃hMin+EM_n²)

≥x

!

≤2 3

^2/3

x^−2/3expn

−1 2x²o

. (3.13)

By condition (A1) and the fact thatEhMin=EM_n²= 1, it is easy to see that 3

2hMin+9

4EM_n²≤3

2(1 +δ²_n) +9 4 ≤ 3

2

1 + 1 16

+9

4 <4.

Therefore, for allx >0, P

Mn≥xp

[M]n, [M]n≥16

≤ P

Mn≥x r3

4[M]n+ 4, [M]n≥16

≤ P

Mn≥x r3

4[M]n+3

2hMin+9

4EM_n², [M]n≥16

≤ P

Mn≥x r3

4[M]n+3

2hMin+9 4EM_n²

= P

Mn≥

r3 2x

r1

2[M]n+hMin+3 2EM_n²

≤ 2

3x^−2/3expn

−3 4x²o as desired.

Lemma 3.9. Assume conditions (A1) and (A2). Then for allρ >0, P |[M]n− hMin| ≥1

≤cρ ε^(2+ρ)/2_n +ε^ρ_n . Proof. Notice that [M]_n− hMin =Pn

i=1(ξ_i²−E[ξ_i²|Fi−1]) is a martingale. For ρ, we distinguish two cases as follows.

Whenρ∈(0,2],by the inequality of [von Bahr and Esseen,1965], it follows that E[|[M]_n− hMin|^(2+ρ)/2] ≤

n

X

i=1

E[|ξ²_i −E[ξ_i²|Fi−1]|^(2+ρ)/2]

≤ c₁

n

X

i=1

E[|ξ_i|^2+ρ]

≤ c2ε^ρ_n,

where the last line follows by conditions (A1) and (A2). Hence, by Markov’s inequality, P |[M]_n− hMin| ≥1

≤ E[|[M]_n− hMin|^(2+ρ)/2]

≤ c2ε^ρ_n,

Whenρ >2,by Rosenthal’s inequality (cf., Theorem 2.12 of [Hall and Heyde,1980]), Lemma3.7, and condition (A2), it follows that

E[|[M]_n− hMi_n|^(2+ρ)/2]

≤cρ,1

EXⁿ

i=1

E[(ξ²_i −E[ξ_i²|Fi−1])²|Fi−1]^(2+ρ)/4 +

n

X

i=1

E|ξ²_i −E[ξ_i²|Fi−1]|^(2+ρ)/2

≤c_ρ,2

EXⁿ

i=1

E[ξ_i⁴|Fi−1]^(2+ρ)/4 +

n

X

i=1

E|ξi|^2+ρ

≤cρ,3 ε^(2+ρ)/2_n +ε^ρ_n

. (3.14)

This completes the proof of the lemma.

Consider the predictable process Ψ(λ) = (Ψk(λ),Fk)k=0,...,n, which is related to the martingaleM as follows:

Ψk(λ) =

k

X

i=1

lnE[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 }),

Ψn(λ)

≤c λ^2+ρε^ρ_n.

In the proofs of Theorems2.2and2.3, we make use of the following assertion, which gives us a rate of convergence in the CLT for the conjugate martingaleY(λ) under the probability measurePλ.

Proposition 3.1. Assume conditions (A1) and (A4). With the convention thatYn(0)/0 = Mn, we have:

[i] Ifρ∈(0,1), then for0≤λ=o(γ_n⁻¹), sup

x

P_λ(Y_n(λ)/λ≤x)−Φ(x) ≤c_ρ

λ^ργ_n^ρ+γ_n^ρ+δ_n . [ii] If ρ= 1, then for0≤λ=o(γ_n⁻¹),

sup

x

Pλ(Yn(λ)/λ≤x)−Φ(x) ≤c

λγn+γn|lnγn|+δn

. Similarly, we have the following Berry-Esseen bound.

Proposition 3.2. Assume conditions (A1), (A2) and (A3). Then for0≤λ=o(max{ε⁻¹_n , κ⁻¹_n }), sup

x

Pλ(Yn(λ)/λ≤x)−Φ(x) ≤cρ

λ^ρ/2γ_n^ρ/2+ε^ρ/(3+ρ)_n +δn

, with the convention thatYn(0)/0 =Mn.

The proofs of Propositions3.1and3.2are much more complicated and we give details in the supplemental article [Fan, Grama, Liu and Shao,2017].

4. Proof of the main results

We start with the proofs of Theorems2.2and2.3, and conclude with the proof of Theorem 2.1. Theorem2.4is an easy consequence of Theorems2.2and2.3

4.1. Proof of Theorem 2.2

Recall that

ζ_i(λ) =λξ_i−1 2λ²ξ²_i. By (3.1), it is easy to see that

n

Sn ≥xp [S]n

o

=n

Mn≥xp [M]n

o⊇

Mn ≥x²+λ²[M]n

2λ

= ⁿ

X

i=1

ζi(λ)≥ x² 2

.

For 0≤λ=o(γ_n⁻¹),according to (3.2), (3.6) and (3.15), we have the following represen- tation:

P

Wn ≥x

= Eλ

h

Zn(λ)⁻¹1

{Sn≥x√

[S]_n}

i

= Eλ

h expn

−

n

X

i=1

ζi(λ) + Ψn(λ)o

1{Mn≥x√

[M]_n}

i

≥ E_λh expn

−Y_n(λ)−B_n(λ) + Ψ_n(λ)o 1{Pn

i=1ζ_i(λ)≥^x₂²}

i

= Eλ

h expn

−Yn(λ)−Bn(λ) + Ψn(λ)o 1_{Y

n(λ)≥^x₂²−Bn(λ)}

i . Using Lemmas3.5,3.4and3.10, we get

P

Wn≥x

≥ Eλ

h expn

−Yn(λ)−1

2λ²hMin+c1λ^2+ρε^ρ_no

×1{Yn(λ)≥^x₂²−(¹₂λ²hMin−c1λ^2+ρε^ρ_n)}

i . Condition (A1) implies that

|hMi_n−1| ≤δ²_n, and thus

P

W_n ≥x

≥ E_λh expn

−Y_n(λ)−1

2λ²+c₁λ^2+ρε^ρ_n

(1 +δ²_n)o

×1_{Y

n(λ)≥^x₂²−(¹₂λ²(1−δ²_n)−c1λ^2+ρε^ρ_n)}

i

. (4.1)

Letλ=λ(x) be the largest solution of the following equation 1

2λ²(1−δ²_n)−c1λ^2+ρε^ρ_n= x² 2 . The definition ofλimplies that for 0≤x=o(γ_n⁻¹),

x≤λ≤c₂ x

p1−δ²_n (4.2)

and

λ=x+c3θ0(x^1+ρε^ρ_n+xδ_n²), (4.3) where 0≤θ0≤1. From (4.1), we obtain

P

Wn≥x

≥exp

−1

2λ²+c1λ^2+ρε^ρ_n

(1 +δ²_n)

E_λh

e^−Yn(λ)1_{Y

n(λ)≥0}

i

. (4.4) SettingFn(y) =P_λ(Yn(λ)≤y),we get

P

Wn ≥x

≥exp

−c4 λ²δ²_n+λ^2+ρε^ρ_n

−λ² 2

Z ∞ 0

e^−ydFn(y). (4.5)

By integration by parts, we have the following bound:

Z ∞ 0

e^−ydFn(y)≥ Z ∞

0

e^−ydΦ(y/λ)−2 sup

y

Fn(y)−Φ(y/λ)

. (4.6)

We distinguish two cases according to the values ofρ.

Case 1:ρ∈(0,1). Combining (4.5) and (4.6), by Proposition3.1, we have for 0≤x= o(γ⁻¹_n ),

P

Wn ≥x

≥ exp

−c4 λ²δ_n²+λ^2+ρε^ρ_n

−λ² 2

× Z ∞

0

e^−λydΦ(y)−c_1,ρ

λ^ργ_n^ρ+γ_n^ρ+δ_n

. (4.7)

Because

e^−λ2/2 Z ∞

0

e^−λydΦ(y) = 1−Φ (λ) (4.8)

and 1

1 +λe^−λ2/2≤√ 2π

1−Φ (λ)

, λ≥0, (4.9)

we obtain the following lower bound P Wn ≥x

1−Φ λ ≥ expn

−c4(λ²δ_n²+λ^2+ρε^ρ_n)o

1−c2,ρ(1 +λ)(λ^ργ_n^ρ+γ_n^ρ+δn)

≥ expn

−c3,ρ

λ²δ²_n+λ^2+ρε^ρ_n+ (1 +λ)(λ^ργ_n^ρ+γ_n^ρ+δn)o

, (4.10) for 0≤λ≤_2c¹

2,ρmin{γn^−ρ/(1+ρ), δ_n⁻¹}.

Next, we consider the case of _2c¹

2,ρmin{γ^−ρ/(1+ρ)n , δ⁻¹_n } ≤λ=o(γ_n⁻¹).LetK ≥1 be an absolute constant, whose exact value is chosen later. It is easy to see that

E_λh

e^−Yn(λ)1_{Y

n(λ)≥0}

i ≥ E_λh

e^−Yn(λ)1_{0≤Y

n(λ)≤λKτ}

i

≥ e^−λKτP_λ

0≤Y_n(λ)≤λKτ

, (4.11)

whereτ=λ^ργ^ρ_n+δn.By Proposition3.1, we have P_λ

0≤Yn(λ)≤λKτ

≥ P

0≤ N(0,1)≤Kτ

−c4,ρτ

≥ 1

√2πKτ e^−K2τ2/2−c4,ρτ

≥ 1

3K−c_4,ρ τ.

LettingK≥12c_4,ρ, it follows that P_λ

0≤Y_n(λ)≤λKτ

≥ 1

4Kτ = 1

4Kλ^1+ργ_n^ρ+λδn

λ .

Choosing

K= maxn

12c_4,ρ, 4

√π(2c_2,ρ)^1+ρo and taking into account that _2c¹

2,ρmin{γn^−ρ/(1+ρ), δ⁻¹} ≤λ=o(γ⁻¹_n ), we conclude that P_λ

0≤Y_n(λ)≤λKτ

≥ 1

√πλ.

Because the inequality ^√1_πλe^−λ2/2 ≥1−Φ (λ) is valid for all λ≥1, it follows that for

1

2c_2,ρmin{γn^−ρ/(1+ρ), δ⁻¹} ≤λ=o(γ⁻¹_n ), P_λ

0≤Yn(λ)≤Kτ

≥

1−Φ λ e^λ

2/2

. (4.12)

Combining (4.4), (4.11), and (4.12), we obtain P Wn≥x

1−Φ λ ≥exp

−c5,ρ

λ²δ_n²+λ^2+ρε^ρ_n+ (1 +λ)(λ^ργ_n^ρ+γ_n^ρ+δn)

, (4.13) which is valid for _2c¹

2,ρmin{γn^−ρ/(1+ρ), δ⁻¹} ≤λ=o(γ_n⁻¹).

From (4.10) and (4.13), we get for 0≤λ=o(γ_n⁻¹), P W_n≥x

1−Φ λ ≥exp

−c_6,ρ

λ²δ_n²+λ^2+ρε^ρ_n+ (1 +λ)(λ^ργ_n^ρ+γ_n^ρ+δ_n)

. (4.14) Next, we substitute xforλin the tail of the normal law 1−Φ(λ).By (4.2), (4.3), and (4.9), we get

1≤ R∞

λ exp{−t²/2}dt R∞

x exp{−t²/2}dt ≤ 1 + Rx

λexp{−t²/2}dt R∞

x exp{−t²/2}dt

≤ 1 +c₁x(x−λ) expn

(x²−λ²)/2o

≤ expn

c2(x²δ_n²+x^2+ρε^ρ_n)o

(4.15) and hence

1−Φ λ

= (1−Φ(x)) exp

θ₁c(x^2+ρε^ρ_n+x²δ_n²) . (4.16) Implementing (4.16) in (4.14) and using (4.2), we obtain for 0≤x=o(γ_n⁻¹),

P Wn≥x 1−Φ (x) ≥exp

−c7,ρ

x^2+ρε^ρ_n+x²δ_n²+ (1 +x)(x^ργ_n^ρ+γ_n^ρ+δn) ,

which gives the desired lower bound (2.7).

Case 2:ρ= 1.Using Proposition 3.1withρ= 1,we have for 0≤x=o(γ_n⁻¹), P

Wn≥x

≥ exp

−c1 λ²δ²_n+λ³εn

−λ² 2

× Z ∞

0

e^−λydΦ(y)−c₂

λγ_n+γ_n|lnγ_n|+δ_n ,

that is, the termγ_n^ρ in inequality (4.7) has been replaced byγn|lnγn|.By an argument similar to that ofCase 1, we obtain the desired lower bound (2.8).

4.2. Proof of Theorem 2.3

We first prove Theorem2.3for 1≤x=o(γ_n⁻¹).Observe that P

Wn≥x

= P

Wn≥x,|[M]n− hMin| ≤δn+ 1/(2x) +P

W_n≥x,|[M]_n− hMin|> 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) withλ=x, we have the following representation:

P

Wn ≥x,|[M]n− hMin| ≤δn+ 1/(2x)

=E_xh

Z_n(x)⁻¹1

{Mn≥x√

[M]n,|[M]n−hMin|≤δn+1/(2x)}

i

=Ex

h

e^{−Yn(x)−Bn(x)+Ψn(x)}1

xMn≥x²√

1+[M]n−1,|[M]n−hMin|≤δn+1/(2x)

i . By the inequality

p1 +y≥1 +y/2−y²/2, y≥ −1, condition (A1) and Lemma3.4, we have for 1≤x=o(γ_n⁻¹),

P

Wn≥x , |[M]n− hMin| ≤δn+ 1/(2x)

≤Ex

h expn

−Yn(x)−Bn(x) + Ψn(x)o

×1xM_n−¹₂x²[M]_n+¹₂x²([M]_n−1)²≥¹₂x²,|[M]_n−hMi_n|≤δ_n+1/(2x)

i

≤E_xh expn

−Y_n(x)−B_n(x) + Ψ_n(x)o

×1xMn−¹₂x²[M]n+x²([M]n−hMin)²+x²(1−hMin)²≥¹₂x²,|[M]n−hMin|≤δn+1/(2x)

i

≤E_xh expn

−Y_n(x)−B_n(x) + Ψ_n(x)o

×1Yn(x)≥−x²([M]n−hMin)²−x²δ⁴_n+¹₂x²−Bn(x),|[M]n−hMin|≤δn+1/(2x)

i .