Scholarly article on topic 'Precise Asymptotics in the Law of Iterated Logarithm for Moving Average Process under Dependence'

Precise Asymptotics in the Law of Iterated Logarithm for Moving Average Process under Dependence Academic research paper on "Mathematics"

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Academic research paper on topic "Precise Asymptotics in the Law of Iterated Logarithm for Moving Average Process under Dependence"

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 320932,17 pages doi:10.1155/2011/320932

Research Article

Precise Asymptotics in the Law of Iterated Logarithm for Moving Average Process under Dependence

Jie Li12

1 Wang Yanan Institute for Studies in Economics, Xiamen University, Xiamen 361005, China

2 School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, China

Correspondence should be addressed to Jie Li, lijiezufe@gmail.com

Received 10 November 2010; Revised 2 February 2011; Accepted 3 March 2011

Academic Editor: Ulrich Abel

Copyright © 2011 Jie Li. 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.

Let -to < i < to} be a doubly infinite sequence of identically distributed and ^-mixing random variables, and let {ai, -to < i < to} be an absolutely summable sequence of real numbers. In this paper, we get precise asymptotics in the law of the logarithm for linear process {Xk = £ ai+k^i, k > 1}, which extend Liu and Lin's (2006) result to moving average process under dependence assumption.

1. Introduction and Main Results

Let -to < i < to} be a doubly infinite sequence of identically distributed random variables with zero means and finite variances, and let {air -to < i < to} be an absolutely summable sequence of real numbers. Let

Xk =£ al+klr, k > 1, (1.1)

i=—to

be the moving average process based on -to <i< to}. As usual, we denote Sn = X £=i Xk, n > 1 as the sequence of partial sums.

Under the assumption that -to < i < to} is a sequence of independent identically distributed random variables, many limiting results have been obtained. Ibragimov [1] established the central limit theorem; Burton and Dehling [2] obtained a large deviation principle; Yang [3] established the central limit theorem and the law of the iterated logarithm;

Li et al. [4] obtained the complete convergence result for {Xk, k > 1}.Aswe know, Xk (k > 1) are dependent even if {&, -to < i < to} is a sequence of i.i.d. random variables. Therefore, we introduce the definition of ty-mixing,

ty(m) := supf\P(B | A) - P(B)\, A e?^, P(A) = 0, B 0, m to, (1.2)

k>i L '

where Fa = a(^i, a < i < b). Many limiting results of moving average for ty-mixing have been obtained. For example, Zhang [5] got complete convergence.

Theorem A. Suppose that {&, -to < i < to} is a sequence of identically distributed and ty-mixing random variables with £TO=1 ty1/2(m) < to, and {Xk, k > 1} is defined as (1.1). Let h(x) > 0 (x > 0) be a slowly varying function and 1 < t < 2, r > 1, then = 0 and E\^1\rth(\Yt\) < to imply

^nr-2h(n)P(\S„\ > nl/te) < to, Ve > 0. (1.3)

Li and Zhang [6] achieved precise asymptotics in the law of the iterated logarithm.

Theorem B. Suppose that {&, -to < i < to} is a sequence of identically distributed and ty-mixing random variables with mean zeros and finite variances, £^ ty1/2(m) < to, and 0 < a2 = E£2 + 2Xn=2 < to, E^2(log+\^1\)5-1 < to, for 6 > 0. Suppose that {X,Xk, k > 1} is defined as in (1.1), where {ai, -to <i < to} is a sequence of real number with£ to_to \ai\ < to, then one has

eme26+2J^^gn-P(\Sn\ > V«1^) = 6T++TE\N\26+2, (1.4)

where t =: a Xto=_to ai, N is a standard normal random variable.

On the other hand, since Hsu and Robbins [7] introduced the concept of the complete convergence, there have been extensions in some directions. For the case of i.i.d. random variables, Davis [8] proved XTO=1(logn/n)P( \Sn\ > yjnlogne) < to, for e > 0 if and only if EX1 = 0, EX2 < to. Gut and Spataru [9] gave the precise asymptotics of XTO=2((log n)6/n)P( \Sn\ > \Jn log ne). We know that complete convergence can be derived from complete moment convergence. Liu and Lin [10] introduced a new kind of convergence of STO=2((log n)6-1/n2)E \ Sn\2I{\ Sn\ > yjn log ne}. In this note, we show that the precise asymptotics for the moment convergence hold for moving-average process when {&, -to < i < to} is a strictly stationary ty-mixing sequences. Now, we state the main results.

Theorem 1.1. Suppose that {X,Xk,k > 1} is defined as in (1.1), where {ai, -to < i < to} is a sequence of real number with £to_to \ai\ < to, and {&, -to < i < to} is a sequence of identically

distributed ty-mixing random variables with mean zeros and finite variancesty1/2(m) < to and 0 < a2 = £¿2 + 2 e"=2 E¿l¿k < to, £^2(log+i^1i)6 < to, for 0 < 6 < 1, then one has

- (log n)5-1 f i- ^ T25+2

lim e25£ ^ g/ E\Sn\2l\ \Sn \ >y/n log nTe\ = ~^E\N |25+2, (1.5)

where t =: aXTO-TO ai-

Theorem 1.2. Under the conditions in Theorem 1.1, one has

sm «2* 1 ^^ Esn^\s„\ > vmog^T^ = Ti!+w+^. (I,)

Remark 1.3. In this paper, we generate the results of Liu and Lin [10] to linear process under dependence based on Theorem B by using the technique of dealing with the innovation process in Zhang [5].

We first proceed with some useful lemmas.

Lemma 1.4. Let {X, Xk, k > 1} be defined as in (1.1), and let {¿i, -to < i < to} be a sequence of identically distributed ty-mixing random variables with £¿1 = 0, £¿2 < to, 0 < a2 = £¿2 + 2STO=2 Ehik < to, 2TO=1 ty1/2(2m) < to, then

N(0,1).

The proof is similar to Theorem 1 in [11]. Set An = supx\P(\Sn\ > ^Jnx) - P(\N\ > x)\. From Lemma 1.4, one can get An ^ 0 as n ^ to.

Lemma 1.5 (see [2]). Let £ +=-to ai be an absolutely convergent series of real numbers with a = £+toto ai and k > 1, then

lim — V

n ^ton

Zaj j=i+1

= \a\k

Lemma 1.6 (see [12]). Let {Xi, i > 1} be a sequence of mixing random variables with zero means and finite second moments. Let Sn = £n=i Xi. If exists Cn such that max1<i<nES^ < Cn, then for all q > 2, there exists C = C(q,$(-)) such that

Emax\Si\q < c(Cqn/2 + Emax \Xi\q\ (1.9)

1<i<n 1<i<n

2. Proofs

Proof of Theorem 1.1. Without loss of generality, we assume that t = 1. We have

x (l0g? ES2nl\|Sn| > jn log ne

= e2 |Snl > +

n=2 n n=2

(log n)6-1 ^

2 f _ 2xP(|Sn| > x)dx.

J ^Jn log ne

Set d(e) = exp(Me 2), where M> 1. By Theorem B, we need to show

lim e26 jj (lognn)— | _ 2xP(|Sn| > x)dx = ^ ^ E|N|26+2

e\0 n=2 n

Vn logn

6(6 +1)'

By Proposition 5.1 in [10], we have

(log n)6-1 ^

î\ X[_ 2xp(|N| > =

e\0 n=2 n

Vn logn

Vn/ 6(6 + 1)

1 E|N|26+2.

Hence, Theorem 1.1 will be proved if we show the following two propositions. □

Proposition 2.1. One has

lim e-jlioi^.

e\0 n=2 n2

2xP(|Sn| > x)dx -

J J n log ne

2xp(|N| > —=)dx

^Jn log ne \ Vn/

= 0. (2.4)

Proof. Write

log n)

Vn logn

2xP(|Sn| > x)dx - [ 2xpf N| > -Ç)dx

' J A /n log ne \ V=/

Vn logn

d(e) (log n)c <X V V logn

x J 2(x + e) P^|Sn| > nlogn(x + e)^ - P^|N| > logn(x + e)^

j(e) (log n)

-(An1 + An2 + An3),

Journal of Inequalities and Applications where

A„i = log n

log n\ 2(x + e)

1^v/logn A

1/Viogn a;

P^|Sn| >\Jn logn(x + e)^ - P^|N| >\Jlogn(x + e)^

An2 = logn _ 2(x + e)P( |Sn| > a/nlogn(x + e) )dx,

an/4 v v '

An3 = log^ 2(x + e)p(|N| > \/logn(x + e)^dx.

a/ v v '

Since n < d(e) implies ^/log ne < VM, we have

An1 < C log nAn

log nA

+ e I < c(An/4 + VMA)"

For An3, by Markov's inequality, we get

An3 < Clogn — 3

■h/v^A/ (log n) (x + e)2

From (2.7) and (2.8), we can get

dx < CAlJ4.

d(e)( log n)

lim e26 V

e\0 n=2 n

-(An1 + An3) = 0.

Note that Xn=1 Xk = £to-to £n=1 ak+i¿i = £to-to ani¿i, where a„ = £n=1 ak+i. By Lemma 1.5, we can assume that

J|ani|i < n, f > 1, a = J |ai|< 1.

Set S'n = ETO=-to ani^I{\anM n logn(x + e)}. As £¿1 = 0,by (2.10), we have

(2.10)

|ESn| < C

E ^ ani&lj | ani ¿i | ^ ^/n log n(x + e)

i=-TO ^

< C J] |ani||¿11^¿1| > \Jn logn(x + e)

i=-TO ^

< CnEi¿l |lj aa|¿11 > \Jn log n(x + e)

Journal of Inequalities and Applications

< CnE\^1 \I j\^1\ >\]n log n(x + e)

< C^E^2)1/^p(\&\ > \Jn log n(x + e)))J C

\jlog n (x + e)

(2.11)

So, when x e (1/^/log nAn/4, to),

IES' I Ei2

1 nl < C-—^-2 < e, for n large enough. (2.12)

\fnlogn(x + e) log n( 1/^/ log nAn/4 + e)'

By (2.12), we have

«(logn:: a„2

« p (logn)6(x + e)

n=2-H/ViognA!/4 n

/ ,- \ ( Jn log n (x + e)

P( sup\aniii\ > \Jnlogn(x + eU+ PI S _ ES'n\> -2-

=: H1 + H2.

(2.13)

Set In; = {; e L, 1/(; + 1) < \anj\ < 1/j, j = 1,2,...}, then j Inj = L (referred by [4]). We can get

£#In, < n(k + 1). (2.14)

P jsup\aniii\ >\fn\ogn(x + e)

< ^ W \aniii\ > \Jn log n(x + e)

< EE p{ i¿li>^| n log nj (x + e)\ < ¿(#In; )p{ i¿li >y/n log nj (x + e)

j=1 i<=Inj J j=1 L

<ee (#inj

)pj y/nlognk(x + e) < < yjnlogn(k + 1)(x + e)

j=1 k>j y

< #Inj )pj yjnlognk(x + e) < i¿1i < yjnlogn(k + 1)(x + e) k=1 j=1

<E n(k + 1)pj y/nlognk(x + e) < i¿li <\Jnlogn(k + 1)(x + e) k=1

VnE&|l{i¿li >yjnlogn(x + e)} •y/log n (x + e)

(2.15)

So, we get

fto d<f) (log n) 6-1/2

Hi < CE|&| E^^-

Jl/^/iogn a/ n=2 n1/2

1/v^gnay4 n=2 n1 X E J^k log k(x + e) < <yj(k + 1) log(k + 1)(x + e)jdx

< C f £logk(x + e) < ¿l|< V(k + 1) log(k + 1)(x + e)\dx£ l/2

0 k=2 n=2 n /

(log n)

< CE¿21^ (x + e)-1 g(logk)6-1l{^/k logk(x + e) < |&| < y/(k + 1) log(k + 1)(x + e)}dx

< CE¿2 |log+i¿li - log(x + e)|(6-1)(x + e)-1I{i¿li > (x + e)}dx

< CE£2|log+|£l| - log e|6 < CE£2(log+i£li)6 + CEi\(- log e)6.

(2.16)

Therefore,

lim e26H1 = 0.

8 Journal of Inequalities and Applications

By Lemma 1.6, noting that ^TO=1 ty1/2(m) < to, for q > 2,

d(e) rto (logn)6-q/1

H2 < c£ v V/2 (x + e)1-q n=2J0 n1+q/2

d(e) /• Si,

x \ E(anihfI [\anii1\ < \Jn log n (x + e)

^ \ i=—to ^

+ J] E\anii1\qW \anii1 \ <\Jn log ne

+ J] E\anii1\qW \Jn log ne < \ anii1 \ < \Jn log n(x + e) I fdx

i=_TO ^ > j

=: H21 + H22 + H23. For H21, we have

d(e) rto (logn)6—q/2 / [ /- i\q/2

H21 <E -(x + e)1—^Ei2^ \anii1 \^nlogn(x + e) ) dx

tAh n \ I v J/ (2.19)

< Ce~26M6+1—q/2. Then, for 0 <6 < 1, q > 2, we have

lim lim sup e26H21 = 0. (2.20)

m ^ to e\0

For H22, we decompose it into two parts,

« rTO (logn)6—q/2

i^Jc n1+q/2 - iejnj

d(t) ,-TO Clogn) ^ TO r 1- 1

E J 0 (x + e)1—q EE \ ani\qE\h \qI[ \ anii1 \ <y/n log ne} dx

< ^—q«log n6_q/2

n=2 n1+q/2

j=1 k=0

k=2n+1

=: H221 + H222.

(j+1)n [ 1--1-

+ £ E\i1 \qI Wk log ke < \i1 \ <^J(k + 1) log(k + 1)e

(2.18)

x E(#Inj)j-q{^E\i1\qI[^k log ke < \i1 \ < ^(k + 1) log(k + 1)ej

It is easy to see that

TO TO TO TO

2(#Inj)(; + 1)-q(m + 1)q-1 < £(#Inj)(j + 1)-1 <2 ianii = £ Xianii < n. (2.22)

j=m j=1 i=-TO j=1 iel^

2 (#Inj)j< Cnm-(q-1. (2.23)

Now, we estimate H221, by (2.23),

d(e) (loe n)6-q/2 2n r ---,-

H221 < e2-q£^/-£Ei^I ^klogke < i¿li < V(k + 1) log(k + 1)e

n=2 n k=0 L

d(e) f I--I- 1 d(e) Hoe n)6-q/2

< ^¿^¿1|qI Wk log ke <|&| (k + 1) log(k + 1)e £ -

k=2 ^ J n=[k/2] n

d(e) (log k)6-q/2 r 1--/-1

< e2-qE kq/2-1 E|¿l Vk log ke < i¿li < V(k + 1) log(k + 1)eJ

d(e) r ---

< £ (log k)6-^I Vk log ke < i¿li < V(k + 1) log(k + 1)e k=2

< £ (log k)|log+ i¿li - log e|E2l Vk log ke < |&| < yj(k + 1) log(k + 1)e k=2

< CE¿2(log+i¿li)6 + CE¿2(-loge)6. For H222, we have

H (log n)6-q/2

H222 < e 'n?2 n1+q/2

X e £ (#Inj )j-qE|^|ql{Vk log ke <i¿li <y/ (k + 1) log(k + 1)e

k=2n+1 j>k/n-1

d(e) (loe n)6-q/2 TO r ---

< e2-qEV «L £ yk log ke < i¿li <V (k + 1) log(k + 1)e

«=9 n " k=2n + 1 L

n1-q/2 n=2 n k=2n+1

\6-1 p<-2 1

< £ Gog k)6-1 E¿2^ Vk log ke < |&| < V(k + 1) log(k + 1)e

< E Oog k) |log+i&| - log e|^2l| Vk log ke < |&| < V(k + 1) log(k + 1)ej

< CE£2(log+|£l|)6 + CE£2(-loge)6.

From (2.24) and (2.25), we can get

lim e26H22 = lim e26H221 + lim e26H222 = 0. (2.26)

e\0 e\0 e\0

Finally, q > 2, and we will get

d(e) (logn)6—q/2 [ i-

H23 < cn^/2 E\i1\q^ \ai1\ >yjn log ne

rTO TO f 2n (j+1)n]

(x + e)1—q£ (#In; + X \

J0 j=1 ^k=0 k=2n+1j

x Ijyk log k(x + e) < \i1\ < y (k + 1) log(k + 1)(x + e) fdx

(logn)6—q/2 [ I- 1

< CY^n/i-Ei!^!\ai1\ >^n log ne j

rTO [ I--j d(e) (log n)6_q/2

x J 0 (x + e)1—qI| £ \ < y 2n log(2n)(x + e) jdx + C^ ^n E\i1 \q

f « , k qq 1 A^k logk(x + e) < \i1 \ < J(k + 1) log(k + 1)(x + e)\dx J0 k=2n+1 (x + e)q lv J

< C«EgIlJk log ke < \i1 \ < J(k + 1) log(k + 1)4 «

k=2 ^ J n=2 n

+ CEil (x + e)—1

x £ (log k)6—1I] \/k log k(x + e) < \i1 \ < (k + 1) log(k + 1)(x + e) \dx

< CV (log k)6Eil H Wk log ke < £1! < W(k + 1) log(k + 1)e

+ CE£l \log+\£1\- log(x + e) \(6—1) (x + e)—1I{\£1 \ > (x + e)}dx

< CE£l\log+\£1\ - loge\6 < CE£l(log+\£1\)6 + CE£l(- loge)6,

(2.27)

lim e26Hi3 = 0. (2.28)

Hence, (2.4) can be referred from (2.9), (2.17), (2.20), (2.26), and (2.28). □

Journal of Inequalities and Applications Proposition 2.2. One has

lim e26

(log n)

n=d(e) + 1

Vn losn

2xP(|Sn| > x)dx - f J.

Vn losn

in| > x}dx Vn/

(2.29)

Proof. Consider the following:

(log n)

n=d(e) + 1

[ 2xP(|Sn| > x)dx - [ 2xp(|N| > -Ç)dx

J -y/n log ne J a /n log ne \ Vn /

\/n log n

n=d(e)+1

n=d(e) + 1

=: G1 + G2.

(log n)

| 2(x + e)P^ |N| >yj log n(x + e)

(log n)

j" 2(x + e)P^|Sn| > Vnlogn(x + e)^dx

(2.30)

We first estimate G1, for d > 26, by Markov's inequality,

Gi < E

(log n)

n=d(e) + 1

0 (logn)(e+2)/2(x + e)e+1

< CM6-e/2e-26.

(2.31)

Hence,

lim lim sup e26G1 = 0. (2.32)

Now, we estimate G2. Here, n > Me 2, so

Esm m eh

y/n log n (x + e) (log n)(x + e)2 M

< < e, for M œ. (2.33)

We have

Gi < E

n=d(e) + 1

ilog^! r(x + e)

P^sup\ani£i\ > \Jnlogn(x + e)

+P( \S'n _ ES'n\ >

yjn log n (x + e)

(2.34)

=: G21 + G22.

We estimate G21 first. Similar to the proof of (2.16), we have

G21 < CE\£1\

/"TO TO

0 n=d(e)

(log n)

6—1/2

n=d(e) + 1

x X Ij \k log k(x + e) < £ \ < Y (k + 1) log(k + 1)(x + e) \dx

CE\£1\r « Ay/k log k(x + e) < \£1\^ (k + 1) log(k + 1)(x + e)\dx

0 k=d(e)+1

k (log n) 6—1/2

n=d(e)+1

/" TO TO

CEil ^ Oog n)6—1(x + e)—1

k=d(e) + 1

x I|yk logk(x + e) < \£1\ < Y(k + 1) log(k + 1)(x + e) \dx

\ log\£1 \ — log(x + e) \(6—1) (x + e)—1I{\£1 \ > (x + e) }dx

< CE£l\log+\£1\ - loge\6 < CE£l(log+\£1\)6 + CE£l(- loge)

6 , nrill

(2.35)

lim limsup e26G21 = 0.

M ^ to e\0

Journal of Inequalities and Applications By Lemma 1.6, for q > 2, we have

(tog nf

G22 = ^ ^T"' | (x + e)P

n=d(e) + 1

IS' - ES' I >

I Sn ESn\ >

\Jn log n(x + e)

n=d(e) + 1

(log n)

n1+q/2

x (x + e) J0

1_q ji^E E^l)2I{iani¿ii < \Jn logn(x + e)

^ \ i=—TO L

+ J] Eiani¿l|q^ |ani^| < Vnlogne

i=-TO ^

+ J] Eiani¿liqЦ \Jn log ne < ianiiii <\Jn log n(x + e)

=: G221 + G222 + G223.

For G221, we have

(2.37)

to fTO (log n)6-q/2 q/2 G221 < H -(x + e)1-q(M{iаni¿li < n(x + e)}) dx

n=d(e)+lJ0 n X 7

< Ce2-q £ (logn)S —q/ < CM6+1-q/2e-26.

Z-J fT

n=d(e)+1

(2.38)

Next, turning to G222, it follows that

G222 < e2-q £

(log n)

n=d(e) + l

n1+q/2

X £ (#Inj )j-q i eEi^lf^k log ke < i¿li < ^ (k + 1) log(k + 1)e

j=1 i k=2 ^

(j+1)n

k=2n+1

+ £ Ei¿liqП yjk log ke <i¿li < ^(k + 1) log(k + 1)e

(2.39)

=: G2221 + G2222,

\6—q/2 in

G2221 < Ce2—q « i^n/-^E\£1\q4Vk log ke < \£1 \ < y/(k + 1) log(k + 1)e

(log n) n=d(e) + 1 nq/ k=2

(log n)6—q/l

< Cel-^E\£1\qn^k log ke < \£1 \ <V(k + 1) log(k + 1)e £ q/2

k=2 ^ J n=[k/2] n

< Ce2—(logqfci_1 q E\£1\q4Vk log ke < \£1 \ < V (k + 1) log(k + 1)e

< C^ (log k)6-:1 E£l^ Vk log ke < \£1 \ < V(k + 1) log(k + 1)e

< C£ (log k)_1 \ log+ \£1 \ - log e\6E£2U yjk log ke < \£1 \ < ^(k + 1) log(k + 1)e

< CE£l(log+\£1\)6 + CE£l(-loge)6.

(2.40)

For G2222, it follows that

Giiii < Ce2-q ^

(log n)6-q/l

n1+q/2 n=d(e)+1 n

x « « (#Inj)j-qE\£1\q4Vk log ke < \£1 \ <y/(k + 1) log(k + 1)e|

k=2n+1j>k/n-1 ^ J

to r /- 1-j [k/2] (logn)6-q/2

< Ce2-q £ k1-qE\£1\qN yk log ke < \£1 \ < y/(k + 1) log(k + 1)e £ V

k=d(e) + 1 ^ n=d(e)+1 n

< C 2 (log k)6-1E£l^Vk log ke < \£1 \ <V(k + 1) log(k + 1)e

k=d(e)+1

< C ^ Gog k) ~>g+ \ £1 \ _ log e\6E£ln yk log ke < \£1 \ < yj (k + 1) log(k + 1)e

k=d(e)+1

< CE£l(log+ \£1 \)6 + CE£l(-loge)6.

Finally, q > 2, we have

G223 < C £ ^ q E^l|qi j |a¿l| > Jn log ne

n=d(e)+1 n1+q/2

fTO TO f 2n (j+1)n'

(x + e)1-q£ (#Inj )H£ + X

J0 j=1 I k=0 k=2n+1

x Ij yjk log k(x + e) < < yj(k + 1) log(k + 1)(x + e) j dx

< C £ (lognq/2 q E|^|ql{ |a¿l| > Jn log ne

n=d(e)+1 nq/2

r-OO r ,__A OO (\^„\6-q/2

X T (x + e)1-ql{|&|< ^2nlog(2n)(x + e)\dx + C £ ^^— Ei¿l|q

0 n=d(e)+1 n q/

xT £ T~~771 logk(x + e) < < a/(k + 1) log(k + 1)(x + e)\dx J0 ^=/„+1 (x + e)q I v J

0 k=m+l(x+e)q 1

(log n)

< C £ E¿2^A/k log ke < i^K^/(k + 1) log(k + 1)ej £ ^

k=d(e) + 1 ^ ^ n=d(e)+1 n

+ CE£2 (x + e)

k /1- - --N6-1

x £ (logk)6-1l{Vklogk(x + e) < i¿li < y/(k + 1) log(k + 1)(x + e)\dx

k=d(e) + 1 ^ J

X (log k)E2l{Vk log ke < i¿li < y/(k + 1) log(k + 1)ej

C )6 2

+ CE¿2 |log+i¿li - log(x + e) |(6-1) (x + e)-1I{i¿li > (x + e)}dx

< ^¿2|log+i^i-loge|6 < CE£2(log+|£l|)6 + CE£?(-loge)6.

(2.42)

From (2.38) to (/.4/), we can get

lim lim e26G22 = 0. (2.43)

M —> to e\0

(/./9) can be derived by (/.3/), (2.36), and (2.43). □

Proof of Theorem 1.2. Without loss of generality, we set t = 1. It is easy to see that X ^nwt Es2nI\\Sn \ > Vn loglog ne

n=3 n2logn

= e2V (longllooggnn) 41 s„ 1 * V^g^e} (2-44)

V (log2log „— f _ 2xP( | S„ | > x)dx.

n=3 „2log „ -

n loglog ne

So, we only prove the following two propositions:

X (loglog^p [ Vnlgog-nA = EJN^, (2.45)

n=3 n log n 1 V O o J 6 + 1

eiS£^^ p ^( \ Sn \> = (i.46)

n=3 n2logn Wn loglog ne 6(6 + 1)

The proof of (2.45) can be referred to [6], and the proof of (2.46) is similar to Propositions 2.1 and 2.2. □

Acknowledgments

The author would like to thank the referee for many valuable comments. This research was supported by Humanities and Social Sciences Planning Fund of the Ministry of Education of PRC. (no. 08JA790118)

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