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

International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 253750,15pages

doi:10.1155/2009/253750

Research Article

Convergence Rates for Probabilities of

Moderate Deviations for Multidimensionally

Indexed Random Variables

Dianliang Deng

Department of Mathematics and Statistics, University of Regina, Regina, SK, Canada S4S 0A2

Correspondence should be addressed to Dianliang Deng,deng@uregina.ca

Received 12 March 2009; Revised 23 August 2009; Accepted 27 September 2009

Recommended by Jewgeni Dshalalow

Let{X, Xn;n∈Zd}be a sequence of i.i.d. real-valued random variables, andSn_k≤nX_k,n∈Zd.

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_nbnψ2_anP{|S

n| ≥anφan}, wherean

n1/α1 1 · · ·n

1/αd

d ,bn n

β1 1 · · ·n

βd

d,φandψare taken from a broad class of functions. These results

generalize and improve some results of Li et al.1992and some previous work of Gut1980.

Copyrightq2009 Dianliang Deng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let Zd

be the set of all positive integerd–dimensional lattice points with coordinate-wise partial ordering,≤; that is, for everym m1, . . . , md,n n1, . . . , nd ∈ Zd,m ≤ nif and only ifmi ≤ni,i 1,2, . . . , d, whered ≥1 is a fixed integer.|n|denote

_d

i1niandn → ∞

meansni → ∞, i 1,2, . . . , d. Throughout the paper,{X, Xn, Xn;n ∈ Z, n ∈ Zd}are i.i.d. random variables withEX0 andEX2σ2. LetSnkn1Xk, andSn_k≤nX_k. we define

an n1₁/α1· · ·n1/αd

d , bn n

β1

1 · · ·n

βd

d , 1.1

whereα1, . . . , αd; β1, . . . , βdare real numbers with 0< αi ≤2, βi≥ −1, i1,2, . . . , d. Further,

setαmax{αi; 1≤i≤d}, smax{αiβi1; 1≤i≤d}, r max{αiβi2; 1≤i≤d}, q

Hartman and Wintner1studied 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 logarithmLIL.

Theorem 1.1. EX 0 andEX2_σ2_{<∞if and only if}

lim sup

n→ ∞

Sn

2nlog logn1/2 −lim infn→ ∞

Sn

2nlog logn1/2 σ a.s. 1.2

Afterward, the study of the estimate of the convergence rate in the above relation1.2 has been attracting the attention of various researchers over the last few decades. Darling and Robbins2, Davis3, Gafurov4, and Li5have obtained some good results on the estimate of convergence rate in1.2. The best result is probably the one given by Li et al.6. For easy reference, we restate their result inTheorem 1.2.

Theorem 1.2. Letφtand ψtbe two positive real-valued functions on1,∞such thatφtis

nondecreasing, limt→ ∞φt ∞,andψt Oφtas t → ∞. For t≥0, letσ2t EX2I{|X|<

√

t} −EXI{|X|<√t}2, andσ2

nΔσ2nφ2n. Then the following results are equivalent:

n≥1

ψ2_n

n P

|Sn| ≥n1/2φn

<∞, 1.3

n≥1

ψ2_n

nφnexp − φ2_n

2σ2

n

<∞. 1.4

If, in addition,EX2I{|X|> t}O1/log logtast → ∞, then1.3is equivalent to the following:

n≥1

ψ2_n

nφnexp

−1

2φ

2_n

<∞. 1.5

Note that the above result is the best possible forn.

Further Strassen7introduced 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∈Zd}, which is called the Wichura LIL.

Theorem 1.3. EX 0, EX2_σ2_,andEX2_log|X|d−1_{<∞if and only if}

lim sup |n| → ∞

Sn

2σ2_{|n| log log|n|}1/2

However the analogue ofTheorem 1.2is 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 relation1.6. Towards this end, we consider the equivalence of the following statements:

n

ψ2_|n| |n| P

|Sn| ≥ |n|φ|n|

<∞, 1.7

n

ψ2_|n| |n|φ|n|exp

⎛

⎝₋φ2|n|

2σ2

|n|

⎞

⎠_{<∞, 1.8}

n

ψ2_|n| |n|φ|n|exp

−1

2φ

2_|n|

<∞, 1.9

whereφtandψtare the same as those inTheorem 1.1.

Gut 9 has obtained some equivalent conditions of 1.7 for special functions

φ|n| log|n|1/2, and ψ|n| log|n|. Forφ|n| log log|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 Sp˘ataru 10, Jiang et al. 11, Jiang and Yang 12, and Su 13. However for the general form of functions φx and ψx there is no discussion on the equivalences of1.7and1.8or1.7and1.9for the multidimensionally indexed random variables.

Therefore the aim of the present paper is to discuss the equivalences of1.7and1.8 or of1.7and1.9. In addition, for generalφandψ, we will also consider the equivalence of the following statements:

n

bnψ2anP|Sn| ≥anφan

<∞, _1.10

n

bnψ2_an|n|1/2

anφan exp −

a2_nφ2_an

2|n|σ2

an

<∞, 1.11

n

bnψ2_an|n|1/2

anφan exp −

a2_nφ2_an

2|n|

<∞. 1.12

The equivalence of1.7and1.8 or1.7and1.9 follows from that of1.10and1.11 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. InSection 2, the main results are stated; proofs of which are presented in Section 4. InSection 3, we give some auxiliary results needed in the proofs of the theorems in

2. Main Results

Let{X, Xn;n ∈ Zd}be a sequence of real-valued random variables withEX 0,EX2 1, and letSn

k≤nXk, wheren ∈ Zd.φt, ψtand σ2tare the same as inTheorem 1.1.

For convenience, we use the symbols σ_n2 to denote σ2|n|φ2|n| and σ_a2_n to denote

σ2_anφ2_{an. Let Lx L}

1x log maxe, x and Lkx LLk−1x for k ≥ 2. We

use Lxand logxinterchangeably. We do the same for L2xand log logx. logx stands for

max{1,logx}.

The following is a general result, which improves a number of existing results in the literature.

Theorem 2.1. If E|X|rlog|X|q−1p < ∞and α 2, then 1.10and 1.11 are equivalent. If,

in addition, EX2I{|X| > anφan} O|n|/a2nφ2an,then 1.10 and 1.12 are

equivalent.

By takingan |n|1/2andbn |n|−1fromTheorem 2.1, respectively, we obtain the following Theorems2.2and2.3.

Theorem 2.2. LetE|X|2β2log|X|q1−1 < ∞,where β max{βi,1 ≤ i ≤ d} > −1,and q1

card{i:βiβ}. Then the following results are equivalent:

n

bnψ2|n|P|Sn| ≥ |n|1/2φ|n|

<∞, _2.1

n

bnψ2_|n|

φ|n| exp

⎛

⎝₋φ2|n|

2σ2

|n|1/2

⎞

⎠_{<∞. 2.2}

Moreover, ifEX2_I{|X|>|n|1/2_φ|n|}Oφ−2_{|n|, then2.1is equivalent to}

n

bnψ2|n| φ|n| exp

−1

2φ

2_|n|

<∞. 2.3

Theorem 2.3. Suppose thatα2 andE|X|2log|X|q2−1<∞, whereq2 card{i:αi2},then

the following are equivalent:

n

|n|−1ψ2anP|Sn| ≥anφan

<∞, _2.4

n

ψ2_an|n|−1/2

anφan exp −

a2_nφ2_an

2|n|σ_a2_n

If, in addition,EX2_{I{|X| ≥anφan}O|n|/a}2_nφ2_{an, then2.4is equivalent to}

n

ψ2_an|n|−1/2

anφan exp −

a2_nφ2_an

2|n|

<∞. 2.6

In particular, we obtain the equivalence of1.7and1.9.

Theorem 2.4. LetE|X|2log|X|d−1<∞andd≥2. Then,1.7and1.9are equivalent.

Remarks. iIf r 2, then the condition that EX2_{I{|X| > anφan} O|n|/}

a2_nφ2_{ancan be replaced byEX}2_{I{|X|> anφan} O|n|/a}2_{nlog logan}

inTheorem 2.1. This leads to the equivalence of the following two results:

n

bnψ2_an|n|1/2

anφan exp −

anφ2_an

2|n|σ_a2_n

<∞, 2.7

n

bnψ₁2an|n|1/2

anφ1an exp −

anφ2₁an

2|n|σ_a2_n

<∞, 2.8

where

φ1t ⎧ ⎨ ⎩

2d1log₂t1/2, ifφt≥2d1log₂t1/2,

φt, otherwise,

ψ1t ⎧ ⎨ ⎩

2d1log₂t1/2, ifψt≥2d1log₂t1/2,

ψt, otherwise.

2.9

To see this, we note that

n

bn2d1log₂an1/2|n|1/2

an exp −

an2d1log₂an

2|n|σ2

an

<∞. 2.10

The equivalence of 2.7 and 2.8 follows immediately from 2.10 and σ2

an → 1, φ1t ≤ φt, andψ1t ≤ ψt. Equation2.10 does not converge if r > 2.Therefore,

the assumption EX2_{I{|X| > |n|}1/2_{φn} Oφ−}2_{|n| in Theorem 2.2 cannot be relaxed.}

Under the assumption thatφt Ot−β1/2β_{, EX}2_{I{|X| ≥anφan} Oφ}−2_|n|can

be removed. Similarly,EX2I{|X| ≥anφan}O|n|/a2nφ2ancan be replaced by

EX2_{I{|X| ≥anφan}O|n|/a}2_nlog

2aninTheorem 2.3.

iiInTheorem 2.4,EX2_log|X|d−1 _{<∞impliesEX}2_{I{|X| ≥}√_{t}O1/log logt.}

Fromi,EX2_I{|X|>|n|1/2_φ|n|}Oφ−2_{|n|may be discarded.}

In the previous theorems, we assumed that ψt Oφt. Now we relax this

assumption and obtain the following better result.

Theorem 2.5. Letφtand ψtbe slowly varying functions, and let φt be nondecreasing with

φt → ∞t → ∞.Then the following are equivalent:

EH|X|rlogH|X|q−1pψH|X|<∞,

n

bnψan|n|1/2

anφan exp −

a2_nφ2_an

2|n|σ2_an

<∞,

2.11

n

bnψanP|Sn| ≥anφan

<∞. 2.12

If, in addition,EX2_{I{|X|> anφan}O|n|/a}2_nφ2_{an,then2.12is equivalent to the}

following:

EH|X|rlogH|X|q−1pψH|X|<∞,

n

bnψan|n|1/2

anφan exp −

a2_nφ2_an

2|n|

<∞,

2.13

whereHXis the reverse function oftφtandα, r, q,andpare the same as inTheorem 2.1.

Similar to Theorems 2.2 and 2.3, by choosing specific φn and ψn, we get the

convergence rate for the LIL of random variables with multidimensional indices.

Corollary 2.6. Under the conditions of Theorem 2.5, if r 2 andq > 1, then the following are equivalent:

EX2log|X|q−1loglog|X|γ−1<∞,

EX0, EX21,

2.14

n

bnlog₂anγP

|Sn| ≥an

δlog₂an

<∞ forδ >2q3, 2.15

whereq3card{i:βi1}. In particular, ford >1, the following are equivalent:

EX2log|X|d−1loglog|X|γ−1<∞,

EX0, EX2_1, 2.16

n

|n|−1log₂|n|γP

|Sn| ≥

2d|n|log₂|n|

Let φ1, . . . , φd;ψ1, . . . , ψd be functions satisfying the previous assumptions. We now

consider the equivalence of the following statements:

n d

i1

ψ2

ini

ni

P|Sn| ≥ |n|1/2φ1n1· · ·φdnd

<∞, 2.18

n d

i1

ψ_i2ni

niφini

exp − 1

2σ2

n d

i1

φ_i2ni

<∞, 2.19

n d

i1

ψ2

ini

niφini

exp −1 2

d

i1

φ2_ini

<∞ 2.20

and obtain the following theorem.

Theorem 2.7. IfEX2_log|X|δ _{< ∞forδ > d−1, then2.18and2.19are equivalent. If, in}

addition,EX2I{|X| ≥ |n|1/2d_i1φini}Odi1φ−2ni, then2.18and2.20are equivalent.

3. Auxiliary Results

In this section we give some lemmas that will be of use later. Again{X, Xn;n∈Zd}are i.i.d.

random variables with mean zero.Cdenotes a generic positive constant which varies from line to line. The notationbetween sums and/or integrals will be used to denote that the quantities on either of the sign converge simultaneously andfx ≈ gxmeans that there are constantsC1andC2such that for everyxgreater than somex0, C1gx≤fx≤C2gx.

Lemma 3.1. Let {αi; 1 ≤ i ≤ d},{τi; 1 ≤ i ≤ d} be real numbers with αi ≥ 0, and let

an n1₁/α1· · ·n1/αd

d , tn n τ1

1 · · ·n

τd

d, and sx be a nondecreasing varying function. Define

α max{αi; 1 ≤ i ≤ d}, r max{αiτi1; 1 ≤ i ≤ d}, q card{i : αiτi1 r}, p

card{i:τi1 0}, andτ max{τi; 1≤i≤d}.For eachx≥0, define

g1x

an≥x

tn, g2x

an≥x

tns|n|,

f1x

an≤x

tn, f2x

an≤x

tns|n|.

3.1

One has the following conclusions.

iIfτ1≤0, thenf1x≈logxp, f2x≈sxlogxp.

iiIfτ1>0, thenf1x≈xrlogxpq−1, f2x≈xrlogxqp−1sx.

iiiIfτ1<0, theng1x≈xrlogxq−1, g2x≈xrlogxq−1sx.

This lemma extends Lemma 1.1 of Giang 14. Since the proof is similar to that of Lemma 1.1 in Giang14, the details are omitted.

Lemma 3.2. Let τi ≤ −1,1 ≤ i ≤ d,tn nτ1₁ · · ·nτ_dd, and k card{i : τi −1}. Then

Proof. Without loss of generality, letτ1 · · · τk −1, andτk1 < −1, . . . , τd <−1. For large

|n|,we have

logn1· · ·nk

−β

lognk1· · ·nd

−β_≤

log|n|−β≤logn1· · ·nk

−β

. 3.2

Thus,

n

tnlog|n|−β≈

n1···nk

n1· · ·nk−1

logn1· · ·nk

₋β

. _3.3

The lemma follows from Lemma 5.1 of Gut9.

Remark 3.3. FromLemma 3.2, one may show that _ntnlog|n|−βlog₂|n|r converges for

−∞< r <∞ifβ > k.

The following lemma gives the estimate of the remainder term in the central limit theorem15, page 125, Theorem 15which 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 withEX 0, EX21, andE|X|3<∞.

Then for allx,

|Fnx−Φx| ≤C

E|X|3

√

n1|x|3

, _3.4

whereFnx P1/√nin1Xi≤x,Φx 1/

√

2πx_−∞e−t2_/2

dt.

4. Proofs of Main Results

Proof ofTheorem 2.1. We first show that

n

bnψ2anPSn≥anφan

−Φ −anφan

|n|1/2σan

<∞, 4.1

whereσ2

anσ2a2nφ2an EX2I{|X| ≤anφan} −EXI{|X| ≤anφan}2. For convenience, letHn anφan.Fork≤n,define

We have that

P{Sn> Hn} −P

⎧ ⎨ ⎩

k≤n

X

k> Hn

⎫ ⎬ ⎭

≤ |n|P{|X| ≥Hn}. 4.3

Now, putMn EXI{|X|< Hn}.Then we easily yield that

|Mn|3≤E|X|3I{|X|< Hn},

|n|−1/2Mn−|n|EXI{|X| ≥Hn} −→0 |n| −→ ∞,

1≥σ2_an−→1 |n| −→ ∞.

4.4

By usingLemma 3.4, we have that

P{Sn > Hn} −Φ

"

− Hn

|n|1/2σan #

≤

P{Sn> Hn} −P

⎧ ⎨ ⎩

k≤n

X

k> Hn

⎫ ⎬ ⎭

P

" −

k≤n

X_k−M_k

|n|1/2σan

<−Hn− |n|Mn

|n|1/2σan #

−Φ −Hn− |n|Mn

|n|1/2σan

Φ −Hn− |n|Mn

|n|1/2σan

−Φ − Hn

|n|1/2σan

≤ |n|P{|X| ≥Hn}C E|X|

3_{I{|X|< Hn}|M}

n|3

|n|1/2

1Hn− |n|Mn/|n|1/2σan

3

C |n||Mn|

|n|1/2σan exp

"

−H2n |n|σ2

an #

≤ |n|P{|X| ≥Hn}C |n| H3_nE|X|

3_{I{|X|< Hn}C} |n|

HnE|X|I{|X| ≥Hn}

≤C |n| H3_nE|X|

3_{I{|X|< Hn}C} |n|

HnE|X|I{|X| ≥Hn}.

Hence,

n

bnψ2anPSn≥anφan

−Φ −anφan

|n|1/2σan

≤C

n

bnψ2_an|n|

Hn E|X|I{|X| ≥Hn}C

n

bnψ2_an|n|

H3_n E|X|

3_{I{|X|< Hn}}

≡I1I2.

4.6

ByLemma 3.1and maxβi1−1/αi1≥r/2−1/2≥1/2>0,we set

I1C

n

bnψ2_an|n|

anφan E|X|I

|X| ≥anφan

≤C

n

bnφan|n| an E|X|I

|X| ≥anφan

≤C

∞

i0 ⎛

⎝

i≤an<i1

bnφan|n| an

⎞

⎠_{E|X|I|X| ≥iφi}

≤C

n

⎛

⎝

i≤an<i1

bnφan|n| an

⎞

⎠∞

ji

E|X|Ijφj≤ |X|<j1φj1

≤C

∞

j0

j

i0 ⎛

⎝

i≤an<i1

bnφan|n| an

⎞

⎠_{E|X|Ijφj≤ |X|<j1φj1}

≤C

∞

j0 ⎛

⎝

0≤an<j1

bnφan|n| an

⎞

⎠E|X|Ijφj≤ |X|<j1φj1

≤C

∞

j0

φj1

⎛

⎝

0≤an≤j1

bn|n| an

⎞

⎠E|X|Ijφj≤ |X|<j1φj1

≤C

∞

j0

φj1j1r−1logj1pq−1E|X|Ijφj≤ |X|<j1φj1

≤C

∞

j0

φ2j1j1rlogj1pq−1Pjφj≤ |X|<j1φj1

≤C

∞

j0

φj1j1rlogj1φj1pq−1Pjφj≤ |X|<j1φj1.

Because lim sup_{x→ ∞}φx1/φx< ∞andφx → ∞,the last expression is finite if and only ifE|X|rlog|X|pq−1<∞.

Similarly, because−1≤βi<−1/2, βi2−3/αi<0,we have that

I2C

n

bnψ2_an|n|

H3_n E|X|

3_{I{|X|< Hn}}

≤C

∞

i1 ⎛

⎝

i−1<an≤i

bnψ2_an|n|

a3_nφ3_an ⎞

⎠_E|X|3_{I|X| ≤iφi}

≤C

∞

i1 ⎛

⎝

i−1<an≤i

bnφ2_an|n|

a3_nφ3_an ⎞

⎠i

j1

E|X|3Ij−1φj−1<|X| ≤jφj

≤C

∞

j1

∞

ij

⎛

⎝

i−1<an≤i

bn|n| a3_nφan

⎞

⎠_E|X|3_{Ij−1φj−1<|X| ≤jφj}

≤C

∞

j1 ⎛

⎝

an≥j−1

bn|n| a3_nφan

⎞

⎠E|X|3Ij−1φj−1<|X| ≤jφj

≤C

∞

j1

1

φj−1

⎛

⎝

an≥j−1

bn|n| a3_n

⎞

⎠_E|X|3_{Ij−1φj−1<|X| ≤jφj}

≤C

∞

j1

φ3_jj−1r−3_logjq−1_j3

φj−1 P

j−1φj−1<|X| ≤jφj

≤C

∞

j1

jφjrlogjφjpq−1Pj−1φj−1<|X| ≤jφj. 4.8

The last expression is finite if and only ifE|X|rlog|X|pq−1<∞.

As above, we have that

n

bnψ2_an P

−Sn≥anφan

−Φ −anφan |n|1/2σan

<∞. 4.9

By4.1and4.9, we set

n

bnψ2anP|Sn| ≥anφan

−Φ −anφan |n|1/2σan

Therefore,

n

bnψ2anP|Sn| ≥anφan

<∞ _4.11

is equivalent to

n

bnψ2anΦ −anφan

|n|1/2σan

<∞ 4.12

4.12is also equivalent to the following:

n

bnψ2_an|n|1/2

anφan exp −

a2_nφ2_an

2|n|σ_a2_n

<∞.

4.13

IfE|X|2I{|X|> anφan}O|n|/a2_nφ2_{an,then4.13is equivalent to}

n

bnψ2_an|n|1/2

anφan exp −

a2_nφ2_an

2|n|

<∞. 4.14

The theorem is proved.

The proofs of Theorems2.2,2.3, and2.4are similar to those ofTheorem 2.1and are omitted. Before provingTheorem 2.5, we first give two lemmas.

Lemma 4.1. Letφtandψtbe slowly varying functions at infinity withφtnondecreasing and

φt → ∞.The following are equivalent:

EH|X|rlogH|X|q−1pψH|X|<∞, 4.15

n

bnψan|n|P|X| ≥anφan<∞, 4.16

Proof. Becauseαiβi2≥αi>0,by usingLemma 3.1, we have

n

bnψan|n|P|X| ≥anφan

∞

i1 ⎛

⎝

i≤an<i1

bnψan|n|

⎞

⎠_{P|X| ≥iφi}

∞

i1 ⎛

⎝

i≤an<i1

bnψan|n|

⎞

⎠∞

j1

Pjφj≤ |X|<j1φj1

∞

j1 ⎛

⎝

1≤an<j1

bnψan|n|

⎞

⎠_Pj≤H|X|<j1

∞

j1

ψj1j1rlogj1q−1pPj ≤H|X|<j1.

4.17

The last expression is finite if and only ifEH|X|rlogH|X|q−1pψH|X|<∞.This completes the proof of the lemma.

Lemma 4.2. nbnψanP{|Sn| ≥anφan}<∞implies4.15.

Proof. We first prove the result for symmetric random variables. One may rearrange{X_k, k ≤

n}and obtain{Xj,1≤j≤ |n|}.By Levy inequality, we set

Pmax_k≤nX_k≥anφanP

max

1≤j≤|n|Xj≥anφan

≤2P

⎧ ⎨ ⎩

|n|

j1

Xj

≥anφan

⎫ ⎬ ⎭

2P|Sn| ≥anφan

<∞,

4.18

but

P

"

max

k≤n

X_k ≥anφan

#

1−1−P|X| ≥anφan|n|

≈ |n|P|X| ≥anφan.

4.19

Thus for large|n|,

|n|P|X| ≥anφan≤CP|Sn| ≥anφan

, 4.20

If {Xn, n ∈ Zd} are nonsymmetric random variables, then by using the standard

symmetrization method, it is easy to prove that for some constantC,

|n|P|X| ≥Canφan≤CP|Sn| ≥anφan

, 4.21

which implies that

n

bnψan|n|P|X| ≥Canφan<∞. 4.22

That is,4.15holds.

Proof ofTheorem 2.5. Similar to the proof of Theorem 2.2, and also by Lemma 4.1,

EH|X|rlogH|X|q−1pψH|X|<∞implies that

n

bnψanPSn≥anφan

−Φ −anφan

|n|1/2σan

<∞. 4.23

Therefore, ifEH|X|rlogH|X|q−1p ψH|X|<∞,2.12is equivalent to

n

bnψanΦ −anφan

|n|1/2σan

<∞, 4.24

which is equivalent to

n

bnψan|n|1/2

anφan exp −

a2nφ2an

2|n|σ2

an

<∞. 4.25

On the other hand, 2.12 implies EH|X|r logH|X|q−1p ψH|X| < ∞

by Lemma 4.2. Hence, 2.11 and 2.12 are equivalent. If, in addition, EX2_{I{|X| >}

anφan}O|n|/a2_nφ2_{an,4.24is equivalent to the following:}

n

bnψan|n|1/2

anφan exp −

a2nφ2an

2|n|

<∞. 4.26

Hence,2.12and2.13are equivalent.

Proof ofTheorem 2.7. Like the proof ofTheorem 2.1,EX2_log|X|d−1 _{< ∞and2.17imply}

2.16. On the other hand, if2.16holds, thenEX2_log|X|δ_{<∞implies2.17. The proof is}

Acknowledgments

The author is very grateful for referee’s comments and suggestions which greatly improve the readability and quality of the current paper. The research is partly supported by the Natural Sciences and Engineering Research Council of Canada.

References

1 P. Hartman and A. Wintner, “On the law of the iterated logarithm,” American Journal of Mathematics, vol. 63, pp. 169–176, 1941.

2 D. A. Darling and H. Robbins, “Iterated logarithm inequalities,” Proceedings of the National Academy of

Sciences of the United States of America, vol. 57, pp. 1188–1192, 1967.

3 J. A. Davis, “Convergence rates for the law of the iterated logarithm,” Annals of Mathematical Statistics, vol. 39, pp. 1479–1485, 1968.

4 M. U. Gafurov, “On the estimate of the rate of convergence in the law of iterated logarithm,” in

Probability Theory and Mathematical Statistics, vol. 1021 of Lecture Notes in Mathematics, pp. 137–144,

Springer, Berlin, Germany, 1982.

5 D. L. Li, “Convergence rates of law of iterated logarithm forB-valued random variables,” Science in

China. Series A, vol. 34, no. 4, pp. 395–404, 1991.

6 D. L. Li, X. C. Wang, and M. B. Rao, “Some results on convergence rates for probabilities of moderate deviations for sums of random variables,” International Journal of Mathematics and Mathematical

Sciences, vol. 15, no. 3, pp. 481–497, 1992.

7 V. Strassen, “An invariance principle for the law of the iterated logarithm,” Zeitschrift f ¨ur

Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 3, pp. 211–226, 1964.

8 M. J. Wichura, “Some Strassen-type laws of the iterated logarithm for multiparameter stochastic processes with independent increments,” The Annals of Probability, vol. 1, pp. 272–296, 1973.

9 A. Gut, “Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices,” The Annals of Probability, vol. 8, no. 2, pp. 298–313, 1980.

10 A. Gut and A. Sp˘ataru, “Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables,” Journal of Multivariate Analysis, vol. 86, no. 2, pp. 398–422, 2003.

11 C. W. Jiang, X. R. Yang, and L. X. Zhang, “Precise asymptotics of partial sums for multidimensionally indexed random variables,” Journal of Zhejiang University. Science Edition, vol. 33, no. 6, pp. 625–628, 2006.

12 C. Jiang and X. Yang, “Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables,” Applied Mathematics: A Journal of Chinese Universities, vol. 22, no. 1, pp. 87–94, 2007.

13 K.-L. Su, “Strong limit theorems of Cesaro type means for arrays of orthogonal random elements with multi-dimensional indices in Banach spaces,” Stochastic Analysis and Applications, vol. 22, no. 2, pp. 237–250, 2004.

14 N. V. Giang, “On moments of the supremum of normed partial sums of random variables indexed by Nk_{,” Acta Mathematica Hungarica, vol. 60, no. 1-2, pp. 73–80, 1992.}

15 V. V. Petrov, Sums of Independent Random Variables, vol. 8 of Ergebnisse der Mathematik und ihrer