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

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

Research Article

Precise Asymptotics in the Law of

Iterated Logarithm for Moving Average

Process under Dependence

Jie Li

1, 2

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,[email protected]

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

Academic Editor: Ulrich Abel

Copyrightq2011 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{ξi, −∞< i <∞}be a doubly infinite sequence of identically distributed andφ-mixing random

variables, and let{ai, −∞< i <∞}be an absolutely summable sequence of real numbers. In this

paper, we get precise asymptotics in the law of the logarithm for linear process{Xki∞−∞aikξi,

k ≥1}, which extend Liu and Lin’s2006result to moving average process under dependence assumption.

1. Introduction and Main Results

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

Xk

∞

i−∞

aikξi, k≥1, 1.1

be the moving average process based on{ξi, −∞< i <∞}. As usual, we denoteSn

_n

k1Xk,

n≥1 as the sequence of partial sums.

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Li et al.4obtained the complete convergence result for{Xk, k≥1}. As we know,Xk k≥1 are dependent even if{ξi, −∞ < i <∞}is a sequence of i.i.d. random variables. Therefore, we introduce the definition ofφ-mixing,

φm:sup k≥1

|PB|A−PB|, A∈ Fk_−∞, PA/0, B∈ F∞_k_m−→0, m−→ ∞, 1.2

whereFb

aσξi, a≤i≤b. Many limiting results of moving average forφ-mixing have been obtained. For example, Zhang5got complete convergence.

Theorem A. Suppose that{ξi, −∞< i <∞}is a sequence of identically distributed andφ-mixing random variables with∞_m₁φ1/2_m_<_∞_{, and}_{_X

k, k≥1}is defined as1.1. Lethx>0 x > 0be a slowly varying function and1≤t <2,r≥1, thenEξ10andE|ξ1|rth|Ytt|<∞imply

∞

n1

nr−2hnP|Sn| ≥n1/t

<∞, ∀ >0. 1.3

Li and Zhang6achieved precise asymptotics in the law of the iterated logarithm.

Theorem B. Suppose that{ξi, −∞ < i <∞}is a sequence of identically distributed andφ-mixing random variables with mean zeros and finite variances,∞_m₁φ1/2_m _< _∞_{, and} ₀ _{< σ}2 _Eξ2

1

2n_k₂Eξ1ξk < ∞,Eξ₁2log|ξ1|δ−1 < ∞, forδ > 0. Suppose that{X, Xk, k ≥ 1}is defined as in

1.1, where{ai, −∞< i <∞}is a sequence of real number with∞i−∞|ai|<∞, then one has

lim 0

2δ2∞

n2

logn δ n P

|Sn| ≥

nlognτ

τ2δ2

δ1E|N|

2δ2_, _1.4

whereτ :σ∞i−∞ai,Nis a standard normal random variable.

On the other hand, since Hsu and Robbins7introduced 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 ∞_n₁logn/nP|Sn| ≥

nlogn < ∞, for > 0 if and only if EX1 0, EX21 < ∞. Gut and Sp˘ataru 9 gave the precise asymptotics of

_∞

n2lognδ/nP|Sn| ≥

nlogn. We know that complete convergence can be derived from complete moment convergence. Liu and Lin10introduced a new kind of convergence of ∞_n₂lognδ−1/n2E|Sn|2I{|Sn| ≥

nlogn}. In this note, we show that the precise asymptotics for the moment convergence hold for moving-average process when{ξi, −∞ <

i <∞}is a strictly stationaryφ-mixing sequences. Now, we state the main results.

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distributedφ-mixing random variables with mean zeros and finite variances,∞_m₁φ1/2_m_<_∞_and

0< σ2_Eξ2 12

_n

k2Eξ1ξk<∞,Eξ21log|ξ1|δ<∞, for0< δ≤1, then one has

lim 0

2δ∞

n2

lognδ−1 n2 E|Sn|

2

I

|Sn| ≥

nlognτ

τ2δ2

δ E|N|

2δ2_, _1.5

whereτ :σ∞i−∞ai.

Theorem 1.2. Under the conditions inTheorem 1.1, one has

lim 0

2δ∞

n3

log logn δ−1 n2_log_n ES

2

nI

|Sn| ≥

nlog lognτ

τ2δ2E|N|2δ1

δ . 1.6

Remark 1.3. In this paper, we generate the results of Liu and Lin10to linear process under dependence based on Theorem B by using the technique of dealing with the innovation process in Zhang5.

We first proceed with some useful lemmas.

Lemma 1.4. Let{X, Xk, k ≥ 1}be defined as in1.1, and let{ξi, −∞ < i < ∞}be a sequence of identically distributedφ-mixing random variables withEξ1 0, Eξ12 < ∞,0 < σ2 Eξ21

2∞_k₂Eξ1ξk<∞,∞m1φ1/22m<∞, then

Sn

τ√n

D

−→N0,1. 1.7

The proof is similar to Theorem 1 in11. SetΔn supx|P|Sn| ≥

√

nx−P|N| ≥x|. FromLemma 1.4, one can getΔn → 0 asn → ∞.

Lemma 1.5see2. Let _i∞_−∞ai be an absolutely convergent series of real numbers with a

_∞

i−∞aiandk≥1, then

lim n→ ∞

1

n

∞

i−∞

in

ji1

aj k

|a|k. 1.8

Lemma 1.6see12. Let{Xi, i≥1}be a sequence ofφ-mixing random variables with zero means and finite second moments. LetSn

_n

i1Xi. If existsCnsuch thatmax1≤i≤nES2n ≤Cn, then for all

q≥2, there existsCCq, φ·such that

Emax

1≤i≤n|Si| q_≤_C

Cq/n 2Emax

1≤i≤n|Xi| q

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2. Proofs

Proof ofTheorem 1.1. Without loss of generality, we assume thatτ1. We have

∞

n2

logn δ−1 n2 ES

2

nI

|Sn| ≥

nlogn

2

∞

n2

logn δ n P

|Sn| ≥

nlogn

∞

n2

logn δ−1 n2

_∞

√

nlogn

2xP|Sn| ≥xdx.

2.1

Setd expM−2, whereM >1. By Theorem B, we need to show

lim 0

2δ∞

n2

logn δ−1 n2

_∞

√

nlogn

2xP|Sn| ≥xdx 1

δδ1E|N|

2δ2_. _2.2

By Proposition 5.1 in10, we have

lim 0

∞

n2

logn δ−1 n2

_∞

√

nlogn 2xP

|N| ≥ √x n

dx 1

δδ1E|N|

2δ2_. _2.3

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

Proposition 2.1. One has

lim 0

2δd

n2

logn δ−1 n2

_∞

√

nlogn

2xP|Sn| ≥xdx−

_∞

√

nlogn 2xP

|N| ≥ √x n

dx0. 2.4

Proof. Write

d

n2

logn δ−1 n2

_∞

√

nlogn

2xP|Sn| ≥xdx−

_∞

√

nlogn 2xP

|N| ≥ √x n

dx

≤

d

n2

logn δ−1 n logn

×

_∞

0

2xP

|Sn| ≥

nlognx

−P

|N| ≥

lognxdx

≤d

n2

logn δ−1

n Δn1 Δn2 Δn3,

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where

Δn1logn

₁_/√_log_n_Δ1/4

n

0

2xP

|Sn| ≥

nlognx

−P

|N| ≥

lognxdx,

Δn2logn

_∞

1/√lognΔ1/4

n

2xP

|Sn| ≥

nlognx

dx,

Δn3logn

_∞

1/√lognΔ1/4

n

2xP

|N| ≥

lognx

dx.

2.6

Sincen≤dimplieslogn≤√M, we have

Δn1≤ClognΔn ⎛ ⎜

⎝ 1

lognΔ1n/4

⎞ ⎟ ⎠ 2

≤CΔ1n/4

MΔn 2

. 2.7

ForΔn3, by Markov’s inequality, we get

Δn3≤Clogn

_∞

1/√lognΔ1/4

n

1

logn 3/2x2dx≤CΔ

1/4

n . 2.8

From2.7and2.8, we can get

lim 0

2δ d

n2

logn δ−1

n Δn1 Δn3 0. 2.9

Note thatn_k₁Xk

_∞

i−∞nk1akiξi

_∞

i−∞aniξi, whereani

_n

k1aki. ByLemma 1.5, we can assume that

∞

i−∞

|ani|t≤n, t≥1, a

∞

i−∞

|ai| ≤1. 2.10

SetSn

_∞

i−∞aniξiI{|aniξi| ≤

nlognx}. AsEξ10, by2.10, we have

_ES

n≤C E

∞

i−∞

aniξiI

|aniξi|>

nlognx

≤C

∞

i−∞

|ani|E|ξ1|I

|aniξ1|>

nlognx

≤CnE|ξ1|I

a|ξ1|>

nlognx

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≤CnE|ξ1|I

|ξ1|>

nlognx

≤CnEξ2₁1/2

P

|ξ1|>

nlognx

1/2

≤C

√

nEξ2 1

lognx .

2.11

So, whenx∈1/

lognΔ1n/4,∞,

|ESn|

nlognx

≤C Eξ

2 1

logn1/lognΔ1n/4

2 < , fornlarge enough. 2.12

By2.12, we have

d

n2

logn δ−1 n Δn2

≤d

n2

_∞

1/√lognΔ1/4

n

logn δx n

×

⎡ ⎢ ⎣P

sup

i

|aniξi|>

nlognx

P

⎛ ⎜

⎝Sn−ESn≥

nlognx

2

⎞ ⎟ ⎠ ⎤ ⎥ ⎦dx

:H1H2.

2.13

SetInj {j ∈ L, 1/j1<|anj| ≤1/j, j 1,2, . . .}, then !

j≥1Inj Lreferred by4. We can get

k

j1

#Inj ≤nk1. 2.14

Then,

P

" sup

i

|aniξi|>

nlognx

#

≤ ∞

i−∞

P

|aniξi|>

nlognx

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≤∞

j1

i∈Inj

P

|ξ1| ≥

nlognjx

≤∞

j1

#Inj P

|ξ1| ≥

nlognjx

≤∞

j1

k≥j

#Inj P

nlognkx≤ |ξ1|<

nlognk1x

≤∞

k1 k

j1

#Inj P

nlognkx≤ |ξ1|<

nlognk1x

≤∞

k1

nk1P

nlognkx≤ |ξ1|<

nlognk1x

≤ √

nE|ξ1|I

|ξ1| ≥

nlognx

lognx

.

2.15

So, we get

H1≤CE|ξ1|

_∞

1/√lognΔ1n/4

d

n2

logn δ−1/2 n1/2

×∞

kn

I

klogkx<|ξ1| ≤

k1logk1x

dx

≤C

_∞

0 ∞

k2

E|ξ1|I

klogkx<|ξ1| ≤

k1logk1x

dx

k

n2

logn δ−1/2 n1/2

≤CEξ2 1

_∞

0

x−1

∞

k2

logk δ−1I

klogkx<|ξ1| ≤

k1logk1x

dx

≤CEξ2₁

_∞

0

log|ξ1| −logxδ−1x−1I{|ξ1|>x}dx

≤CEξ21log|ξ1| −logδ≤CEξ12

log|ξ1| δCEξ12

−log δ.

2.16

Therefore,

lim 0

2δ_H

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ByLemma 1.6, noting that∞_m₁φ1/2_m_<_∞_{, for}_{q >}_2,

H2≤C

d

n2

_∞

0

logn δ−q/2 n1q/2 x

1−q

×

⎧ ⎨ ⎩

_∞

i−∞

Eaniξ12I

|aniξ1| ≤

nlognx

q/2

∞

i−∞

E|aniξ1|qI

|aniξ1| ≤

nlogn

∞

i−∞

E|aniξ1|qI

nlogn <|aniξ1| ≤

nlognx

⎫_⎬

⎭dx

:H21H22H23.

2.18

ForH21, we have

H21≤

d

n2

_∞

0

logn δ−q/2 n x

1−q

Eξ21I

|aniξ1| ≤

nlognx

q/2

dx

≤C−2δMδ1−q/2.

2.19

Then, for 0< δ≤1,q >2, we have

lim

M→ ∞lim sup₀ 2δ_H

210. 2.20

ForH22, we decompose it into two parts,

H22≤

d

n2

_∞

0

logn δ−q/2 n1q/2 x

1−q∞

j1

i∈Inj

|ani|qE|ξ1|qI

|aniξ1| ≤

nlogn

dx

≤2−q

d

n2

logn δ−q/2 n1q/2

×∞

j1

#Inj j−q ⎧ ⎨ ⎩

2n

k0

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

j1n

k2n1

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

⎫_⎬

⎭

:H221H222.

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It is easy to see that

∞

jm

#Inj j1 −qm1q−1≤

∞

j1

#Inj j1 −1 ≤

∞

i−∞

|ani|

∞

j1

i∈Inj

|ani| ≤n. 2.22

So,

∞

jm

#Inj j−q ≤Cnm−q−1. 2.23

Now, we estimateH221, by2.23,

H221≤2−q

d

n2

logn δ−q/2 nq/2

2n

k0

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

≤2−q

d

k2

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

d

nk/2

≤2−q

d

k2

logk δ−q/2 kq/2−1 E|ξ1|

q_I_k_log_k_{≤ |}_ξ

1|<

k1logk1

≤d

k2

logk δ−1Eξ2₁I

klogk≤ |ξ1|<

k1logk1

≤

d

k2

logk −1log|ξ1| −logδEξ₁2I

klogk≤ |ξ1|<

k1logk1

≤CEξ₁2log|ξ1| δCEξ12

−log δ.

2.24

ForH222, we have

H222≤2−q

d

n2

logn δ−q/2 n1q/2

× ∞

k2n1

j≥k/n−1

#Inj j−qE|ξ1|qI

klogk≤ |ξ1|<

k1logk1

≤2−q

d

n2

logn δ−q/2 n1−q/2

∞

k2n1

k1−qE|ξ1|qI

klogk≤ |ξ1|<

k1logk1

≤

d

k2

logk δ−1Eξ₁2I

klogk≤ |ξ1|<

k1logk1

≤d

k2

klogk≤ |ξ1|<

k1logk1

≤CEξ₁2log|ξ1| δCEξ2₁−logδ.

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From2.24and2.25, we can get

lim 0

2δ_H

22 lim

0 2δ_H

221lim

0 2δ_H

2220. 2.26

Finally,q >2, and we will get

H23 ≤C

d

n2

logn δ−q/2 n1q/2 E|ξ1|

q

I

|aξ 1|>

nlogn

×

_∞

0

x1−q

∞

j1

#Inj j−q ⎧ ⎨ ⎩

2n

k0

j1n

k2n1 ⎫ ⎬ ⎭

×I

klogkx<|ξ1| ≤

k1logk1x

dx

≤C

d

n2

logn δ−q/2 nq/2 E|ξ1|

q

I

|aξ 1|>

nlogn

×

_∞

0

x1−qI

|ξ1| ≤

2nlog2nx

dxC

d

n2

logn δ−q/2 n1−q/2 E|ξ1|

q

×

_∞

0 ∞

k2n1

k1−q

xq−1I

klogkx<|ξ1| ≤

k1logk1x

dx

≤C

∞

k2

Eξ2₁I

klogk <|ξ1| ≤

k1logk1

k

n2

logn δ−1 n

CEξ₁2

_∞

0

x−1

×∞

k4

logk δ−1I

klogkx<|ξ1| ≤

k1logk1x

dx

≤C

∞

k2

logk δEξ2 1I

klogk <|ξ1| ≤

k1logk1

CEξ₁2

_∞

0

≤CEξ21log|ξ1| −logδ≤CEξ21

log|ξ1| δCEξ12

−log δ,

2.27

then

lim 0

2δ_H

230. 2.28

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Proposition 2.2. One has

lim 0

2δ ∞

nd1

logn δ−1 n2

_∞

√

nlogn

2xP|Sn| ≥xdx−

_∞

√

nlogn 2xP

|N| ≥√x n

dx0.

2.29

Proof. Consider the following:

∞

nd1

logn δ−1 n2

_∞

√

nlogn

2xP|Sn| ≥xdx−

_∞

√

nlogn 2xP

|N| ≥ √x n

dx

≤ ∞

nd1

logn δ n

_∞

0

2xP

|N| ≥

lognx

∞

nd1

logn δ n

_∞

0

2xP

|Sn| ≥

nlognx

dx

:G1G2.

2.30

We first estimateG1, forθ >2δ, by Markov’s inequality,

G1≤ ∞

nd1

logn δ n

_∞

0

1

logn θ2/2xθ1dx

≤CMδ−θ/2−2δ.

2.31

Hence,

lim

M→ ∞lim sup₀ 2δ_G

10. 2.32

Now, we estimateG2. Here,n > M−2, so

|ESn|

nlognx

< Eξ

2 1

logn x2 < Eξ2

1

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We have

G2≤ ∞

nd1

logn δ n

_∞

0

x

⎡ ⎢ ⎣P

sup

i

|aniξi|>

nlognx

P

⎛ ⎜

⎝Sn−ESn≥

nlognx

2

⎞ ⎟ ⎠ ⎤ ⎥ ⎦dx

:G21G22.

2.34

We estimateG21first. Similar to the proof of2.16, we have

G21≤CE|ξ1|

_∞

0 ∞

nd1

logn δ−1/2 n1/2

×∞

kn

I

klogkx<|ξ1| ≤

k1logk1x

dx

≤CE|ξ1|

_∞

0 ∞

kd1

I

klogkx<|ξ1| ≤

k1logk1x

dx

× k

nd1

logn δ−1/2 n1/2

≤CEξ12

_∞

0 ∞

kd1

logn δ−1x−1

×I

klogkx<|ξ1| ≤

k1logk1x

dx

≤CEξ₁2

_∞

0

≤CEξ₁2log|ξ1| −logδ≤CEξ12

log|ξ1| δCEξ12

−log δ,

2.35

then

lim

M→ ∞lim sup₀ 2δ_G

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ByLemma 1.6, forq >2, we have

G22 ∞

nd1

logn δ n

_∞

0

xP

⎧ ⎪ ⎨ ⎪ ⎩S

n−ESn≥

nlognx

2

⎫ ⎪ ⎬ ⎪ ⎭dx

≤ ∞

nd1

logn δ−q/2 n1q/2

×

_∞

0

x1−q

⎧ ⎨ ⎩

_∞

i−∞

Eaniξ12I

|aniξi| ≤

nlognx

q/2

∞

i−∞

E|aniξ1|qI

|aniξi| ≤

nlogn

∞

i−∞

E|aniξ1|qI

nlogn <|aniξi| ≤

nlognx

⎫_⎬

⎭dx

:G221G222G223.

2.37

ForG221, we have

G221≤ ∞

nd1

_∞

0

lognδ−q/2 n x

1−q_Eξ2

1I{|aniξ1| ≤nx} q/2

dx

≤C2−q

∞

nd1

logn δ−q/2 n ≤CM

δ1−q/2−2δ_.

2.38

Next, turning toG222, it follows that

G222≤2−q ∞

nd1

logn δ−q/2 n1q/2

×∞

j1

#Inj j−q ⎧ ⎨ ⎩

2n

k2

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

j1n

k2n1

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

⎫_⎬

⎭

:G2221G2222,

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then

G2221≤C2−q ∞

nd1

2n

k2

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

≤C2−q

∞

k2

E|ξ1|qI

klogk≤ |ξ1|<

k1logk1

_∞

nk/2

≤C2−q

∞

k2

logk δ−q/2 kq/2−1 E|ξ1|

q

I

klogk≤ |ξ1|<

k1logk1

≤C

∞

k2

logk δ−1Eξ12I

klogk≤ |ξ1|<

k1logk1

≤C

∞

k2

logk −1log|ξ1| −logδEξ12I

klogk≤ |ξ1|<

k1logk1

≤CEξ2₁log|ξ1| δCEξ21

−log δ.

2.40

ForG2222, it follows that

G2222≤C2−q ∞

nd1

logn δ−q/2 n1q/2

× ∞

k2n1

j≥k/n−1

#Inj j−qE|ξ1|qI

klogk≤ |ξ1|<

k1logk1

≤C2−q

∞

kd1

k1−qE|ξ1|qI

klogk≤ |ξ1|<

k1logk1

k/2

nd1

logn δ−q/2 n1−q/2

≤C

∞

kd1

logk δ−1Eξ12I

klogk≤ |ξ1|<

k1logk1

≤C

∞

kd1

klogk≤ |ξ1|<

k1logk1

≤CEξ2₁log|ξ1| δCEξ21

−log δ.

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Finally,q >2, we have

G223≤C ∞

nd1

logn δ−q/2 n1q/2 E|ξ1|

q

I

|aξ 1|>

nlogn

×

_∞

0

x1−q

∞

j1

#Inj j−q ⎧ ⎨ ⎩

2n

k0

j1n

k2n1 ⎫ ⎬ ⎭

×I

klogkx<|ξ1| ≤

k1logk1x

dx

≤C

∞

nd1

logn δ−q/2 nq/2 E|ξ1|

q

I

|aξ 1|>

nlogn

×

_∞

0

x1−qI

|ξ1| ≤

2nlog2nx

dxC

∞

nd1

logn δ−q/2 n1−q/2 E|ξ1|

q

×

_∞

0 ∞

k2n1

k1−q

xq−1I

klogkx<|ξ1| ≤

k1logk1x

dx

≤C

∞

kd1

Eξ2₁I

klogk <|ξ1| ≤

k1logk1

k

nd1

logn δ−1 n

CEξ₁2

_∞

0

x−1

× ∞

kd1

logk δ−1I

klogkx<|ξ1| ≤

k1logk1x

dx

≤C

∞

k2

logk δEξ2 1I

klogk <|ξ1| ≤

k1logk1

CEξ₁2

_∞

0

≤CEξ2

1log|ξ1| −logδ≤CEξ21

log|ξ1| δCEξ21

−log δ.

2.42

From2.38to2.42, we can get

lim M→ ∞lim0

2δ_G

22 0. 2.43

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Proof ofTheorem 1.2. Without loss of generality, we setτ1. It is easy to see that

∞

n3

log logn δ−1 n2_log_n ES

2

nI

|Sn| ≥

nlog logn

2

∞

n3

log logn δ nlogn P

|Sn| ≥

nlog logn

∞

n3

log logn δ−1 n2_log_n

_∞

√

nlog logn

2xP|Sn| ≥xdx.

2.44

So, we only prove the following two propositions:

2δ2

∞

n3

log logn δ nlogn P

|Sn| ≥

nlog logn

E|N|2δ1

δ1 , 2.45

2δ

∞

n3

log logn δ−1 n2_log_n

_∞

√

nlog logn

2xP|Sn| ≥xdx E|N|

2δ1

δδ1 . 2.46

The proof of2.45can be referred to6, and the proof of2.46is similar to Propositions2.1 and2.2.

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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