# Convergence Rates for Probabilities of Moderate Deviations for Multidimensionally Indexed Random Variables

Authors: Dianliang Deng
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

Let{X,Xn¯;n¯∈Z+d} be a sequence of i.i.d. real-valued random
variables, and
Sn¯=∑k¯≤n¯Xk¯, n¯∈Z+d. Convergence rates of moderate deviations are derived; that is, the rates of
convergence to zero of certain tail probabilities of the partial
sums are determined. For example, we obtain equivalent
conditions for the convergence of the series
∑n¯b(n¯)ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}, where a(n¯)=n11/α1⋯nd1/αd, b(n¯)=n1β1⋯ndβd, ϕ and ψ are taken from a broad class of functions. These results
generalize and improve some results of Li et al. (1992)
and some previous work of Gut (1980).


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## Body

## 1. Introduction

LetZ+d be the set of all positive integer d–dimensional  lattice points with coordinate-wise partial ordering, ≤; that is, for every m¯=(m1,…,md),  n¯=(n1,…,nd)∈Z+d, m¯≤n¯ if and only if mi≤ni, i=1,2,…,d, where d≥1 is a fixed integer. |n¯| denote ∏i=1dni and n¯→∞ means ni→∞,i=1,2,…,d. Throughout the paper, {X,Xn,Xn¯;n∈Z+,n¯∈Z+d} are i.i.d. random variables with EX=0 and EX2=σ2. Let Sn=∑k=1nXk, and Sn¯=∑k¯≤n¯Xk¯. we define(1.1)a(n¯)=n11/α1⋯nd1/αd,b(n¯)=n1β1⋯ndβd,
where α1,…,αd;β1,…,βd are real numbers with 0<αi≤2,βi≥-1,i=1,2,…,d. Further, set α=max{αi;1≤i≤d},s=max{αi(βi+1);1≤i≤d},r=max{αi(βi+2);1≤i≤d},q=card{i:αi(βi+2)=r}, and p=card{i:αi(βi+2)=1}.Hartman and Wintner [1] studied the fundamental strong laws of classic Probability Theory for i.i.d random variables {X,Xn;n∈Z+} and obtained the following Hartman-Wintner law of the iterated logarithm (LIL).Theorem 1.1.
EX=0 and EX2=σ2<∞ if and only if
(1.2)limsupn→∞Sn(2nloglogn)1/2=-liminfn→∞Sn(2nloglogn)1/2=σa.s.Afterward, the study of the estimate of the convergence rate in the above relation (1.2) has been attracting the attention of various researchers over the last few decades. Darling and Robbins [2], Davis [3], Gafurov [4], and Li [5] have obtained some good results on the estimate of convergence rate in (1.2). The best result is probably the one given by Li et al. [6]. For easy reference, we restate their result in Theorem 1.2.Theorem 1.2.
Letϕ(t) and ψ(t) be two positive real-valued functions on [1,∞) such that ϕ(t) is nondecreasing, limt→∞ϕ(t)=∞, and ψ(t)=O(ϕ(t)) as t→∞. For t≥0, let σ2(t)=EX2I{|X|<t}-(EXI{|X|<t})2, and σn2=Δσ2(nϕ2(n)). Then the following results are equivalent:
(1.3)∑n≥1ψ2(n)nP{|Sn|≥n1/2ϕ(n)}<∞,(1.4)∑n≥1ψ2(n)nϕ(n)exp(-ϕ2(n)2σn2)<∞.
If, in addition, EX2I{|X|>t}=O(1/loglogt) as t→∞, then (1.3) is equivalent to the following:
(1.5)∑n≥1ψ2(n)nϕ(n)exp(-12ϕ2(n))<∞.Note that the above result is the best possible forn.Further Strassen [7] introduced an almost sure invariance principle for the Brownian motion and obtained Strassen law of the iterated logarithm. In [8], Wichura generalized Strassen laws of the iterated logarithm for the stochastic processes with multidimensional indices and derived the following version of LIL for multidimensionally indexed i.i.d. random variables {X,Xn¯;n¯∈Z+d}, which is called the Wichura LIL.Theorem 1.3.
EX=0,EX2=σ2, and EX2(log+|X|)d-1<+∞ if and only if
(1.6)limsup|n¯|→∞Sn¯(2σ2|n¯|loglog|n¯|)1/2=da.s.However the analogue of Theorem1.2 is not yet available in the situation, where the i.i.d. random variables are multidimensionally indexed. This motivates us to consider the estimate of convergence rate in the relation (1.6). Towards this end, we consider the equivalence of the following statements:(1.7)∑n¯ψ2(|n¯|)|n¯|P{|Sn¯|≥|n¯|ϕ(|n¯|)}<∞,(1.8)∑n¯ψ2(|n¯|)|n¯|ϕ(|n¯|)exp(-ϕ2(|n¯|)2σ|n¯|2)<∞,(1.9)∑n¯ψ2(|n¯|)|n¯|ϕ(|n¯|)exp(-12ϕ2(|n¯|))<∞,
where ϕ(t) and ψ(t) are the same as those in Theorem 1.1.Gut [9] has obtained some equivalent conditions of (1.7) for special functions ϕ(|n¯|)=ϵ(log|n¯|)1/2, and ψ(|n¯|)=(log|n¯|). For ϕ(|n¯|)=loglog|n¯|, and ψ(|n¯|)=log|n¯|, he also obtained sufficient conditions of (1.7). Recently, many researchers focus on the precise asymptotics in some strong limit theorems for multidimensionally indexed random variables (see Gut and Spătaru [10], Jiang et al. [11], Jiang and Yang [12], and Su [13]). However for the general form of functions ϕ(x) and ψ(x) there is no discussion on the equivalences of (1.7) and (1.8) or (1.7) and (1.9) for the multidimensionally indexed random variables.Therefore the aim of the present paper is to discuss the equivalences of (1.7) and (1.8) or of (1.7) and (1.9). In addition, for general ϕ and ψ, we will also consider the equivalence of the following statements:(1.10)∑n¯b(n¯)ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}<∞,(1.11)∑n¯b(n¯)ψ2(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|σa(n¯)2)<∞,(1.12)∑n¯b(n¯)ψ2(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|)<∞.
The equivalence of (1.7) and (1.8) (or (1.7) and (1.9) ) follows from that of (1.10) and (1.11) (or (1.10) and (1.12) ). Thus the main results of this paper not only generalize the result of Li et al. [6], but also improve the theorems of Gut [9]. The remainder of the paper is organized as follows. In Section 2, the main results are stated; proofs of which are presented in Section 4. In Section 3, we give some auxiliary results needed in the proofs of the theorems in Section 2.


## 2. Main Results

Let{X,Xn¯;n¯∈Z+d} be a sequence of real-valued random variables with EX=0, EX2=1, and let Sn¯=∑k¯≤n¯Xk¯, where n¯∈Z+d. ϕ(t),ψ(t) and σ2(t) are the same as in Theorem 1.1. For convenience, we use the symbols σn¯2 to denote σ2(|n¯|ϕ2(|n¯|)) and σa(n¯)2 to denote σ2(a(n¯)ϕ2(a(n¯))). Let Lx=L1x=logmax(e,x) and Lkx=L(Lk-1x) for k≥2. We use Lx and logx interchangeably. We do the same for L2x and loglogx.   log+x stands for max{1,logx}.The following is a general result, which improves a number of existing results in the literature.Theorem 2.1.
IfE|X|r(log+|X|)q-1+p<∞ and α=2, then (1.10) and (1.11) are equivalent. If, in addition, EX2I{|X|>a(n¯)ϕ(a(n¯))}=O(|n¯|/(a2(n¯)ϕ2(a(n¯)))), then (1.10) and (1.12) are equivalent.By takinga(n¯)=|n¯|1/2 and b(n¯)=|n¯|-1 from Theorem 2.1, respectively, we obtain the following Theorems 2.2 and 2.3.Theorem 2.2.
LetE|X|2(β+2)(log+|X|)q1-1<∞, where β=max{βi,1≤i≤d}>-1, and q1=card{i:βi=β}. Then the following results are equivalent:
(2.1)∑n¯b(n¯)ψ2(|n¯|)P{|Sn¯|≥|n¯|1/2ϕ(|n¯|)}<∞,(2.2)∑n¯b(n¯)ψ2(|n¯|)ϕ(|n¯|)exp(-ϕ2(|n¯|)2σ|n¯|1/22)<∞.
Moreover, if EX2I{|X|>|n¯|1/2ϕ(|n¯|)}=O(ϕ-2(|n¯|)), then (2.1) is equivalent to
(2.3)∑n¯b(n¯)ψ2(|n¯|)ϕ(|n¯|)exp(-12ϕ2(|n¯|))<∞.Theorem 2.3.
Suppose thatα=2 and E|X|2(log+|X|)q2-1<+∞, where q2=card{i:αi=2}, then the following are equivalent:
(2.4)∑n¯|n¯|-1ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}<∞,(2.5)∑n¯ψ2(a(n¯))|n¯|-1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|σa(n¯)2)<∞.
If, in addition, EX2I{|X|≥a(n¯)ϕ(a(n¯))}=O(|n¯|/a2(n¯)ϕ2(a(n¯))), then (2.4) is equivalent to
(2.6)∑n¯ψ2(a(n¯))|n¯|-1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|)<∞.In particular, we obtain the equivalence of (1.7) and (1.9).Theorem 2.4.
LetE|X|2(log+|X|)d-1<∞ and d≥2. Then, (1.7) and (1.9) are equivalent.Remark 2 s.
(i)Ifr=2, then the condition that EX2I{|X|>a(n¯)ϕ(a(n¯))}=O(|n¯|/(a2(n¯)ϕ2(a(n¯)))) can be replaced by EX2I{|X|>a(n¯)ϕ(a(n¯))}=O(|n¯|/(a2(n¯)logloga(n¯))) in Theorem 2.1. This leads to the equivalence of the following two results:
(2.7)∑n¯b(n¯)ψ2(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a(n¯)ϕ2(a(n¯))2|n¯|σa(n¯)2)<∞,(2.8)∑n¯b(n¯)ψ12(a(n¯))|n¯|1/2a(n¯)ϕ1(a(n¯))exp(-a(n¯)ϕ12(a(n¯))2|n¯|σa(n¯)2)<∞,
where
(2.9)ϕ1(t)={2(d+1)(log2t)1/2,ifϕ(t)≥2(d+1)(log2t)1/2,ϕ(t),otherwise,ψ1(t)={2(d+1)(log2t)1/2,ifψ(t)≥2(d+1)(log2t)1/2,ψ(t),otherwise.
To see this, we note that
(2.10)∑n¯b(n¯)(2(d+1)log2a(n¯))1/2|n¯|1/2a(n¯)exp(-a(n¯)2(d+1)log2a(n¯)2|n¯|σa(n¯)2)<∞.
The equivalence of (2.7) and (2.8) follows immediately from (2.10) and σa(n¯)2→1,ϕ1(t)≤ϕ(t), and ψ1(t)≤ψ(t). Equation (2.10) does not converge if r>2. Therefore, the assumption EX2I{|X|>|n¯|1/2ϕ(n¯)}=O(ϕ-2(|n¯|)) in Theorem 2.2 cannot be relaxed. Under the assumption that ϕ(t)=O(t-(β+1)/2β),EX2I{|X|≥a(n¯)ϕ(a(n¯))}=O(ϕ-2(|n¯|)) can be removed. Similarly, EX2I{|X|≥a(n¯)ϕ(a(n¯))}=O(|n¯|/a2(n¯)ϕ2(a(n¯))) can be replaced by EX2I{|X|≥a(n¯)ϕ(a(n¯))}=O(|n¯|/(a2(n¯)log2a(n¯))) in Theorem 2.3.
(ii) In Theorem2.4,  EX2(log+|X|)d-1<+∞ implies EX2I{|X|≥t}=O(1/(loglogt). From (i), EX2I{|X|>|n¯|1/2ϕ(|n¯|)}=O(ϕ-2(|n¯|)) may be discarded.
(iii) Ifd=1, we can obtain Theorem  2.1 of Li et al. [6] from Theorem 2.1.In the previous theorems, we assumed thatψ(t)=O(ϕ(t)). Now we relax this assumption and obtain the following better result.Theorem 2.5.
Letϕ(t) and ψ(t) be slowly varying functions, and let ϕ(t) be nondecreasing with ϕ(t)→∞(t→∞). Then the following are equivalent:
(2.11)E(H(|X|))r(log+H(|X|))q-1+pψ(H(|X|))<∞,∑n¯b(n¯)ψ(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|σa(n¯)2)<∞,(2.12)∑n̅b(n̅)ψ(a(n̅))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}<∞.
If, in addition, EX2I{|X|>a(n¯)ϕ(a(n¯))}=O(|n¯|/a2(n¯)ϕ2(a(n¯))), then (2.12) is equivalent to the following:
(2.13)E(H(|X|))r(log+H(|X|))q-1+pψ(H(|X|))<∞,∑n¯b(n¯)ψ(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|)<∞,
where H(X) is the reverse function of tϕ(t) and α,r,q, and p are the same as in Theorem 2.1.Similar to Theorems2.2 and 2.3, by choosing specific ϕ(n¯) and ψ(n¯), we get the convergence rate for the LIL of random variables with multidimensional indices.Corollary 2.6.
Under the conditions of Theorem2.5, if r=2 and q>1, then the following are equivalent:
(2.14)EX2(log+|X|)q-1(log+log+|X|)γ-1<∞,EX=0,EX2=1,(2.15)∑n¯b(n¯)(log2a(n¯))γP{|Sn¯|≥a(n¯)δlog2a(n¯)}<∞forδ>2q3,
where q3=card{i:βi=1}. In particular, for d>1, the following are equivalent:
(2.16)EX2(log+|X|)d-1(log+log+|X|)γ-1<∞,EX=0,EX2=1,(2.17)∑n¯|n¯|-1(log2|n¯|)γP{|Sn¯|≥2(d+ϵ)|n¯|log2|n¯|}<∞for eachϵ>0.Letϕ1,…,ϕd;ψ1,…,ψd be functions satisfying the previous assumptions. We now consider the equivalence of the following statements:(2.18)∑n¯(∏i=1dψi2(ni)ni)P{|Sn¯|≥|n¯|1/2ϕ1(n1)⋯ϕd(nd)}<∞,(2.19)∑n¯(∏i=1dψi2(ni)niϕi(ni))exp(-12σn¯2∏i=1dϕi2(ni))<∞,(2.20)∑n¯(∏i=1dψi2(ni)niϕi(ni))exp(-12∏i=1dϕi2(ni))<∞
and obtain the following theorem.Theorem 2.7.
IfEX2(log+|X|)δ<∞ for δ>d-1, then (2.18) and (2.19) are equivalent. If, in addition, EX2I{|X|≥|n¯|1/2∏i=1dϕi(ni)}=O(∏i=1dϕ-2(ni)), then (2.18) and (2.20) are equivalent.


## 3. Auxiliary Results

In this section we give some lemmas that will be of use later. Again{X,Xn¯;n¯∈Z+d} are i.i.d. random variables with mean zero. C denotes a generic positive constant which varies from line to line. The notation ≃ between sums and/or integrals will be used to denote that the quantities on either of the sign converge simultaneously and f(x)≈g(x) means that there are constants C1 and C2 such that for every x greater than some x0,C1g(x)≤f(x)≤C2g(x).Lemma 3.1.
Let{αi;1≤i≤d},{τi;1≤i≤d} be real numbers with αi≥0, and let a(n¯)=n11/α1⋯nd1/αd,t(n¯)=n1τ1⋯ndτd, and  s(x) be a nondecreasing varying function. Define α=max{αi;1≤i≤d},r=max{αi(τi+1);1≤i≤d},q=card{i:αi(τi+1)=r},p=card{i:(τi+1)=0}, and τ=max{τi;1≤i≤d}. For each x≥0, define
(3.1)g1(x)=∑a(n¯)≥xt(n¯),g2(x)=∑a(n¯)≥xt(n¯)s(|n¯|),f1(x)=∑a(n¯)≤xt(n¯),f2(x)=∑a(n¯)≤xt(n¯)s(|n¯|).
One has the following conclusions.(i)
Ifτ+1≤0, then f1(x)≈(log+x)p,f2(x)≈s(x)(log+x)p.(ii)
Ifτ+1>0, then f1(x)≈xr(log+x)p+q-1,f2(x)≈xr(log+x)q+p-1s(x).(iii)
Ifτ+1<0, then g1(x)≈xr(log+x)q-1,g2(x)≈xr(log+x)q-1s(x).This lemma extends Lemma  1.1 of Giang [14]. Since the proof is similar to that of Lemma  1.1 in Giang [14], the details are omitted.Lemma 3.2.
Letτi≤-1,(1≤i≤d), t(n¯)=n1τ1⋯ndτd, and k=card{i:τi=-1}. Then ∑n¯t(n¯)(log|n¯|)-β converges if β>k and diverges if β≤k.Proof.
Without loss of generality, letτ1=⋯=τk=-1,  and τk+1<-1,…,τd<-1. For large |n¯|, we have
(3.2)(logn1⋯nk)-β(lognk+1⋯nd)-β≤(log|n¯|)-β≤(logn1⋯nk)-β.
Thus,
(3.3)∑n¯t(n¯)(log|n¯|)-β≈∑n1⋯nk(n1⋯nk)-1(logn1⋯nk)-β.
The lemma follows from Lemma  5.1 of Gut [9].Remark 3.3.
From Lemma3.2, one may show that ∑n¯t(n¯)(log|n¯|)-β(log2|n¯|)r converges for -∞<r<∞ if β>k.The following lemma gives the estimate of the remainder term in the central limit theorem [15, page  125, Theorem  15] which plays an important role in the proofs of theorems in this paper.Lemma 3.4.
Let{X,Xn;n≥1} be i.i.d. random variables with EX=0,EX2=1, and E|X|3<+∞. Then for all x,
(3.4)|Fn(x)-Φ(x)|≤CE|X|3n(1+|x|3),
where Fn(x)=P((1/n)∑i=1nXi≤x),Φ(x)=(1/2π)∫-∞xe-t2/2dt.


## 4. Proofs of Main Results

Proof of Theorem2.1.
We first show that(4.1)∑n¯b(n¯)ψ2(a(n¯))|P{Sn¯≥a(n¯)ϕ(a(n¯))}-Φ(-a(n¯)ϕ(a(n¯))|n¯|1/2σa(n¯))|<∞,
where σa(n¯)2=σ2(a2(n¯)ϕ2(a(n¯)))=EX2I{|X|≤a(n¯)ϕ(a(n¯))}-(EXI{|X|≤a(n¯)ϕ(a(n¯))})2.
For convenience, letH(n¯)=a(n¯)ϕ(a(n¯)). For k¯≤n¯, define
(4.2)Xk′=XkI{|Xk|<H(n¯)}.
We have that
(4.3)|P{Sn¯>H(n¯)}-P{∑k¯≤n¯Xk¯′>H(n¯)}|≤|n¯|P{|X|≥H(n¯)}.
Now, put Mn¯=EXI{|X|<H(n¯)}. Then we easily yield that
(4.4)|Mn¯|3≤E|X|3I{|X|<H(n¯)},|n¯|-1/2Mn¯=-|n¯|EXI{|X|≥H(n¯)}→0(|n¯|→∞),1≥σa(n¯)2→1(|n¯|→∞).
By using Lemma 3.4, we have that
(4.5)|P{Sn¯>H(n¯)}-Φ{-H(n¯)|n¯|1/2σa(n¯)}|≤|P{Sn¯>H(n¯)}-P{∑k¯≤n¯Xk¯′>H(n¯)}|+|P{-∑k¯≤n¯(Xk¯-Mk¯)|n¯|1/2σa(n¯)<-H(n¯)-|n¯|Mn¯|n¯|1/2σa(n¯)}-Φ(-H(n¯)-|n¯|Mn¯|n¯|1/2σa(n¯))|+|Φ(-H(n¯)-|n¯|Mn¯|n¯|1/2σa(n¯))-Φ(-H(n¯)|n¯|1/2σa(n¯))|≤|n¯|P{|X|≥H(n¯)}+CE|X|3I{|X|<H(n¯)}+|Mn¯|3|n¯|1/2(1+|(H(n¯)-|n¯|Mn¯)/|n¯|1/2σa(n¯)|3)+C|n¯||Mn¯||n¯|1/2σa(n¯)exp{-H2(n¯)|n¯|σa(n¯)2}≤|n¯|P{|X|≥H(n¯)}+C|n¯|H3(n¯)E|X|3I{|X|<H(n¯)}+C|n¯|H(n¯)E|X|I{|X|≥H(n¯)}≤C|n¯|H3(n¯)E|X|3I{|X|<H(n¯)}+C|n¯|H(n¯)E|X|I{|X|≥H(n¯)}.
Hence,
(4.6)∑n¯b(n¯)ψ2(a(n¯))|P{Sn¯≥a(n¯)ϕ(a(n¯))}-Φ(-a(n¯)ϕ(a(n¯))|n¯|1/2σa(n¯))|≤C∑n¯b(n¯)ψ2(a(n¯))|n¯|H(n¯)E|X|I{|X|≥H(n¯)}+C∑n¯b(n¯)ψ2(a(n¯))|n¯|H3(n¯)E|X|3I{|X|<H(n¯)}≡I1+I2.
By Lemma 3.1 and max(βi+1-1/αi+1)≥r/2-1/2≥1/2>0, we set
(4.7)I1=C∑n¯b(n¯)ψ2(a(n¯))|n¯|a(n¯)ϕ(a(n¯))E|X|I{|X|≥a(n¯)ϕ(a(n¯))}≤C∑n¯b(n¯)ϕ(a(n¯))|n¯|a(n¯)E|X|I{|X|≥a(n¯)ϕ(a(n¯))}≤C∑i=0∞(∑i≤a(n¯)<i+1b(n¯)ϕ(a(n¯))|n¯|a(n¯))E|X|I{|X|≥iϕ(i)}≤C∑n¯(∑i≤a(n¯)<i+1b(n¯)ϕ(a(n¯))|n¯|a(n¯))∑j=i∞E|X|I{jϕ(j)≤|X|<(j+1)ϕ(j+1)}≤C∑j=0∞∑i=0j(∑i≤a(n¯)<i+1b(n¯)ϕ(a(n¯))|n¯|a(n¯))E|X|I{jϕ(j)≤|X|<(j+1)ϕ(j+1)}≤C∑j=0∞(∑0≤a(n¯)<j+1b(n¯)ϕ(a(n¯))|n¯|a(n¯))E|X|I{jϕ(j)≤|X|<(j+1)ϕ(j+1)}≤C∑j=0∞ϕ(j+1)(∑0≤a(n¯)≤j+1b(n¯)|n¯|a(n¯))E|X|I{jϕ(j)≤|X|<(j+1)ϕ(j+1)}≤C∑j=0∞ϕ(j+1)(j+1)r-1(logj+1)p+q-1E|X|I{jϕ(j)≤|X|<(j+1)ϕ(j+1)}≤C∑j=0∞ϕ2(j+1)(j+1)r(logj+1)p+q-1P{jϕ(j)≤|X|<(j+1)ϕ(j+1)}≤C∑j=0∞(ϕ(j+1)(j+1))r(log(j+1)ϕ(j+1))p+q-1P{jϕ(j)≤|X|<(j+1)ϕ(j+1)}.
Because limsupx→∞ϕ(x+1)/ϕ(x)<+∞ and ϕ(x)→∞,  the last expression is finite if and only if  E|X|r(log+|X|)p+q-1<+∞.
Similarly, because-1≤βi<-1/2,βi+2-3/αi<0, we have that
(4.8)I2=C∑n¯b(n¯)ψ2(a(n¯))|n¯|H3(n¯)E|X|3I{|X|<H(n¯)}≤C∑i=1∞(∑i-1<a(n¯)≤ib(n¯)ψ2(a(n¯))|n¯|a3(n¯)ϕ3(a(n¯)))E|X|3I{|X|≤iϕ(i)}≤C∑i=1∞(∑i-1<a(n¯)≤ib(n¯)ϕ2(a(n¯))|n¯|a3(n¯)ϕ3(a(n¯)))∑j=1iE|X|3I{(j-1)ϕ(j-1)<|X|≤jϕ(j)}≤C∑j=1∞∑i=j∞(∑i-1<a(n¯)≤ib(n¯)|n¯|a3(n¯)ϕ(a(n¯)))E|X|3I{(j-1)ϕ(j-1)<|X|≤jϕ(j)}≤C∑j=1∞(∑a(n¯)≥j-1b(n¯)|n¯|a3(n¯)ϕ(a(n¯)))E|X|3I{(j-1)ϕ(j-1)<|X|≤jϕ(j)}≤C∑j=1∞1ϕ(j-1)(∑a(n¯)≥j-1b(n¯)|n¯|a3(n¯))E|X|3I{(j-1)ϕ(j-1)<|X|≤jϕ(j)}≤C∑j=1∞ϕ3(j)(j-1)r-3(logj)q-1j3ϕ(j-1)P{(j-1)ϕ(j-1)<|X|≤jϕ(j)}≤C∑j=1∞(jϕ(j))r(logjϕ(j))p+q-1P{(j-1)ϕ(j-1)<|X|≤jϕ(j)}.
The last expression is finite if and only if E|X|r(log+|X|)p+q-1<+∞.
As above, we have that(4.9)∑n¯b(n¯)ψ2(a(n¯))|P{-Sn¯≥a(n¯)ϕ(a(n¯))}-Φ(-a(n¯)ϕ(a(n¯))|n¯|1/2σa(n¯))|<∞.
By (4.1) and (4.9), we set
(4.10)∑n¯b(n¯)ψ2(a(n¯))|P{|Sn¯|≥a(n¯)ϕ(a(n¯))}-Φ(-a(n¯)ϕ(a(n¯))|n¯|1/2σa(n¯))|<∞.
Therefore,(4.11)∑n¯b(n¯)ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}<∞
is equivalent to
(4.12)∑n¯b(n¯)ψ2(a(n¯))Φ(-a(n¯)ϕ(a(n¯))|n¯|1/2σa(n¯))<∞
(4.12)is also equivalent to the following:
(4.13)∑n¯b(n¯)ψ2(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|σa(n¯)2)<∞.
If E|X|2I{|X|>a(n¯)ϕ(a(n¯))}=O(|n¯|/a2(n¯)ϕ2(a(n¯))), then (4.13) is equivalent to
(4.14)∑n¯b(n¯)ψ2(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|)<∞.
The theorem is proved.The proofs of Theorems2.2, 2.3, and 2.4 are similar to those of Theorem 2.1 and are omitted. Before proving Theorem 2.5, we first give two lemmas.Lemma 4.1.
Letϕ(t) and ψ(t) be slowly varying functions at infinity with ϕ(t) nondecreasing and ϕ(t)→∞. The following are equivalent:
(4.15)E(H(|X|))r(log+H(|X|))q-1+pψ(H(|X|))<∞,(4.16)∑n¯b(n¯)ψ(a(n¯))|n¯|P{|X|≥a(n¯)ϕ(a(n¯))}<∞,
where H(x) is the reverse function of xϕ(x).Proof.
Becauseαi(βi+2)≥αi>0, by using Lemma 3.1, we have
(4.17)∑n¯b(n¯)ψ(a(n¯))|n¯|P{|X|≥a(n¯)ϕ(a(n¯))}≃∑i=1∞(∑i≤a(n¯)<i+1b(n¯)ψ(a(n¯))|n¯|)P{|X|≥iϕ(i)}=∑i=1∞(∑i≤a(n¯)<i+1b(n¯)ψ(a(n¯))|n¯|)∑j=1∞P{jϕ(j)≤|X|<(j+1)ϕ(j+1)}=∑j=1∞(∑1≤a(n¯)<j+1b(n¯)ψ(a(n¯))|n¯|)P{j≤H(|X|)<(j+1)}≃∑j=1∞ψ(j+1)(j+1)r(log(j+1))q-1+pP{j≤H(|X|)<(j+1)}.
The last expression is finite if and only if E(H(|X|))r(log+H(|X|))q-1+pψ(H(|X|))<+∞. This completes the proof of the lemma.Lemma 4.2.
∑n¯b(n¯)ψ(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}<∞ implies (4.15).Proof.
We first prove the result for symmetric random variables. One may rearrange{Xk¯,k¯≤n¯} and obtain {Xj,1≤j≤|n¯|}. By Levy inequality, we set
(4.18)P{maxk¯≤n¯|Xk¯|≥a(n¯)ϕ(a(n¯))}=P{max1≤j≤|n¯||Xj|≥a(n¯)ϕ(a(n¯))}≤2P{|∑j=1|n¯|Xj|≥a(n¯)ϕ(a(n¯))}=2P{|Sn¯|≥a(n¯)ϕ(a(n¯))}<∞,
but
(4.19)P{maxk¯≤n¯|Xk¯|≥a(n¯)ϕ(a(n¯))}=1-(1-P{|X|≥a(n¯)ϕ(a(n¯))})|n¯|≈|n¯|P{|X|≥a(n¯)ϕ(a(n¯))}.
Thus for large |n¯|,(4.20)|n¯|P{|X|≥a(n¯)ϕ(an¯)}≤CP{|Sn¯|≥a(n¯)ϕ(a(n¯))},
equation (4.15) follows from Lemma 4.1.
If{Xn¯,n¯∈Z+d} are nonsymmetric random variables, then by using the standard symmetrization method, it is easy to prove that for some constant C,(4.21)|n¯|P{|X|≥Ca(n¯)ϕ(an¯)}≤CP{|Sn¯|≥a(n¯)ϕ(a(n¯))},
which implies  that
(4.22)∑n¯b(n¯)ψ(a(n¯))|n¯|P{|X|≥Ca(n¯)ϕ(a(n¯))}<∞.
That is, (4.15) holds.Proof of Theorem2.5.
Similar to the proof of Theorem2.2, by Lemma 4.1, E(H(|X|))r(log+H(|X|))q-1+pψ(H(|X|))<∞ implies that
(4.23)∑n¯b(n¯)ψ(a(n¯))|P{Sn¯≥a(n¯)ϕ(a(n¯))}-Φ(-a(n¯)ϕ(a(n¯))|n¯|1/2σa(n¯))|<∞.
Therefore, if E(H(|X|))r(log+H(|X|))q-1+pψ(H(|X|))<∞, (2.12) is equivalent to
(4.24)∑n¯b(n¯)ψ(a(n¯))Φ(-a(n¯)ϕ(a(n¯))|n¯|1/2σa(n¯))<∞,
which is equivalent to
(4.25)∑n¯b(n¯)ψ(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|σa(n¯)2)<∞.
On the other hand, (2.12) implies E(H(|X|))r(log+H(|X|))q-1+pψ(H(|X|))<∞ by Lemma 4.2. Hence, (2.11) and (2.12) are equivalent. If, in addition, EX2I{|X|>a(n¯)ϕ(a(n¯))}=O(|n¯|/a2(n¯)ϕ2(a(n¯))), (4.24) is equivalent to the following:
(4.26)∑n¯b(n¯)ψ(a(n¯))|n¯|1/2a(n¯)ϕ(a(n¯))exp(-a2(n¯)ϕ2(a(n¯))2|n¯|)<∞.
Hence, (2.12) and (2.13) are equivalent.Proof of Theorem2.7.
Like the proof of Theorem2.1, EX2(log+|X|)d-1<+∞ and (2.17) imply (2.16). On the other hand, if (2.16) holds, then EX2(log+|X|)δ<∞ implies (2.17). The proof is complete.



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*Source: 253750-2009-11-11.xml*