-Mixing Random Variables

Volume 2008, Article ID 140548,10pages doi:10.1155/2008/140548

Research Article

On Chung-Teicher Type Strong Law of Large Numbers for ρ

-Mixing Random Variables

Anna Kuczmaszewska

Department of Applied Mathematics, Lublin University of Technology, Nadbystrzycka 38 D, 20-618 Lublin, Poland

Correspondence should be addressed to Anna Kuczmaszewska,[email protected] Received 23 December 2007; Revised 05 March 2008; Accepted 20 March 2008

Recommended by Stevo Stevic

In this paper the classical strong laws of large number of Kolmogorov, Chung, and Teicher for independent random variables were generalized on the case ofρ^∗-mixing sequence. The main result was applied to obtain a Marcinkiewicz SLLN.

Copyrightq2008 Anna Kuczmaszewska. 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{Xn, n ≥1}be a sequence of random variables defined on the probability spaceΩ,F, P with value in a real spaceRand letSn_n

i1Xi. We say that the sequence{Xn, n≥1}satisfies the strong law of large numbersSLLNif there exists some increasing sequence{bn, n ≥1}

and some sequence{an, n≥1}such that _n

i1 Xi−ai

bn −→0 a.s. asn−→ ∞. 1.1

In this paper, we consider the strong law of large numbers for sequences of dependent random variables which are said to beρ^∗-mixing. To introduce the concept ofρ^∗-mixing sequence, we need the maximal correlation coefficient defined as follows:

ρ^∗k sup

S,T

sup

X∈L²F_S, Y∈L²F_T

covX, Y

√VarX·VarY

, 1.2

whereS,Tare the finite subsets of positive integers such that distS, T infx∈S,y∈T|x−y| ≥k andFW is theσ-field generated by the random variables{Xi, i∈W⊂N}.

Definition 1.1. A sequence of random variables{Xn, n ≥1}is said to be aρ^∗-mixing sequence if

n→∞limρ^∗n<1. 1.3

ρ^∗-mixing random variables were investigated by many authors. Various moment inequalities for sums and maximum of partial sums can be found in papers by Bradley 1, Bryc and Smole ´nski2, Peligrad3, Peligrad and Gut4, and Utev and Peligrad5.

These inequalities are used in many papers concerning the problems of invariance principle Utev and Peligrad5, CLTPeligrad3, or complete convergence for some stochastically dominated sequence ofρ^∗-mixing random variablesCai6, and for an array of rowwiseρ^∗- mixing random variablesZhu7. They will be also important in our further consideration.

The aim of this paper is to give some sufficient conditions for SLLN for a sequence of ρ^∗-mixing random variables without assumptions of identical distribution and stochastical domination. The result presented in this paper is obtained by using the maximal type inequality and the following strong law of large numbers proved by Fazekas and Klesov8.

Theorem 1.2Fazekas and Klesov8. Let{bn, n≥1}be a nondecreasing, unbounded sequence of positive numbers. Let{αn, n ≥1}be nonnegative numbers. Letrbe a fixed positive number. Assume that for eachn≥1:

E max1≤i≤nSi

r≤ⁿ

i1

αi. 1.4

If

∞ i1

αi

b^r_i <∞, 1.5

then

n→∞lim Sn

bn 0 a.s. 1.6

Using this theorem, we are going to show that classical Kolmogorov, Chung, and Teicher’s strong law of large numbers for independent random variables{Xn, n≥1}Chung 9and Teicher10can be generalized to the case ofρ^∗-mixing sequences.

In our further consideration, we need the following result.

Lemma 1.3Utev and Peligrad5. Let {Xn, n ≥ 1}be a ρ^∗-mixing sequence with EXn 0, E|Xn|^q<∞,n≥1,q≥2. Then, there exists a positive constantcsuch that

Emax

1≤k≤n

k i1

Xi

q

≤c _n

i1

EXi^q _n

i1

EX_i² q/2

, ∀n≥1. 1.7

LetCdenote a constant which is not necessary the same in its each appearance.

2. The main result

Let{ϕn, n ≥1}be a sequence of nonnegative, even, continuous and nondecreasing on0,∞ functionsϕn:R→R with lim_x→∞ϕnx ∞and such that for alln≥1 and 1< p≤2:

a ϕnx/x or b ϕnx/x, ϕnx/x^p asx−→ ∞. 2.1 Theorem 2.1. Let{Xn, n≥1}be a sequence ofρ^∗-mixing random variables and let{bn, n≥1}be an increasing sequence of positive real numbers. Let 1< p≤2.

Assume that{ϕn, n≥1}satisfiesain2.1with A ^∞

i2

b^−p_i E

ϕ^p_iXi ϕ^p_i

bi

ϕ^p_iXi _i−1

j1

b^p_jE

ϕ^p_jXj ϕ^p_j

bj

ϕ^p_jXj

<∞, 2.2

B ^∞

n1

PXn≥an

<∞, 2.3

for some sequence{an, n≥1}of positive numbers such that C ^∞

n1

E ϕ^2pn

minXn, an ϕ^2pn

bn

ϕ^2pnXn

<∞, 2.4

or{ϕn, n≥1}satisfiesbin2.1with A₁ ^∞

i2

b^−p_i E

ϕiXi ϕi

bi

ϕiXi _i−1

j1

b^p_jE

ϕjXj ϕj

bj

ϕjXj

<∞, 2.5

andBfor some sequence{an, n≥1}of positive numbers such that C₁ ^∞

n1

E ϕ²_n

minXn, an ϕ²n

bn

ϕ²nXn

<∞. 2.6

Then,

b⁻¹_n ⁿ

i1

Xi−E

XiIXi< bi

−→0 a.s.n−→ ∞. 2.7

Proof. LetX_iXiI|Xi|< bi,X^∗_i X_i−EX_i,S_nn

i1X_i^∗, andSn_n

i1Xi. Then,

∞ n1

P

Xn/Xn

^∞

n1

PXn≥bn

≤^∞

n1

P

2ϕ^2r_nXn≥ϕ^2r_n bn

ϕ^2r_nXn

≤C^∞

n1

E

ϕ^2r_nXn ϕ^2rn

bn

ϕ^2rnXn·I

|Xn| ≥an E

ϕ^2r_nXn ϕ^2rn

bn

ϕ^2rnXn·IXn< an

≤C _∞

n1

PXn≥an ^∞

n1

E ϕ^2r_n

minXn, an ϕ^2rn

bn

ϕ^2rnXn

<∞

2.8

forr pin the caseaorr 1 in the caseb. Hence, the sequences{Xn, n ≥ 1}and{X_n, n≥1}are equivalent in Khinchin’s sense.

Thus, we need only to show that Sn

bn −→0 a.s. n−→ ∞. 2.9

ByLemma 1.3, forq4, we have

E

max1≤i≤nS_i ⁴≤C _n

i1

EX_i^∗4 _n

i1

EX_i^∗2 ₂

≤C

2 n

i1

EX_i^∗4 2 n

i2

EX_i^∗2i−1

j1

EX_j^∗2

≤C

EX₁^{4 n}

i2

EX_i⁴ EX_i²ⁱ⁻¹

j1

EX_j²

.

2.10 ByTheorem 1.2applied with

α1EX₁⁴, αiEX_i⁴ EX_i²ⁱ⁻¹

j1

EX_j² fori2,3, . . . , n, 2.11

we see that in order to show2.9it is enough to prove that ∞

i1

αi

b⁴_i <∞, 2.12

which holds if

∞ i1

EX_i⁴

b⁴_i <∞, ^∞

i2

EX_i² b⁴_i

i−1 j1

EX_j²<∞. 2.13

PutI:_∞

i1P|Xi| ≥ai. Then, we have ∞

i1

EX_i⁴ b⁴_i ≤^∞

i1

E

XiIXi<min

ai, bi4

b⁴_i I≤^∞

i1

EXi^2pIXi<min ai, bi

b^2p_i I.

2.14 Moreover, we note that

a

b·Ia≤b≤2· a

a b ∀a, b >0. 2.15

Hence, in caseawe get, byBandC, ∞

i1

EX_i⁴ b⁴_i ≤^∞

i1

E

ϕ^2p_i Xi ϕ^2p_i

bi IXi<min

ai, bi I≤2

∞ i1

E ϕ^2p_i

minXi, ai ϕ^2p_i

bi

ϕ^2p_i Xi

I <∞, 2.16

while in casebwe get, byBandC1, ∞

i1

EX_i⁴ b_i⁴ ≤2

∞ i1

E ϕ²_i

minXi, ai ϕ²_i

bi

ϕ²_iXi

I <∞. 2.17

Using the fact that

b⁻⁴_i E X_i²

E X_j²

≤b^−2p_i EX_i^pEX_j^p ∀j < i 2.18 both in either casea

∞ i2

EX_i² b⁴_i

i−1 j1

EX_j²

≤^∞

i2

EX_i^p b^2p_i

i−1 j1

EX_j^p≤4 ∞

i2

b^−p_i E

ϕ^p_iXi ϕ^p_i

bi

ϕ^p_iXi _i−1

j1

b^p_jE

ϕ^p_jXj ϕ^p_j

bj

ϕ^p_jXj

<∞, 2.19 or caseb

∞ i2

EX_i² b_i⁴

i−1 j1EX_j²

≤^∞

i2

EX_i^p b^2p_i

i−1 j1

EX_j^p≤4 ∞

i2

b^−p_i E

ϕiXi ϕi

bi

ϕiXi _i−1

j1

b^p_jE

ϕjXj ϕj

bj

ϕjXj

∞.

2.20 Thus, we have established2.9and consequently2.7.

Corollary 2.2. Let{Xn, n≥1}be a sequence ofρ^∗-mixing random variables satisfying the condition ∞

i1

E Xi^rt i^r Xi^rt

<∞ 2.21

with rp for 0< t <1 and all 1< p≤2 , or r2/t for 1≤t <2 . Then,

n^−1/tn

i1

Xi−E

XiIXi< i^1/t

−→0 a.s. asn−→ ∞, 2.22 Proof. Letanbnn^1/tfor any 0< t <2. Then, the assumptionBofTheorem 2.1is fulfilled.

Indeed we see B:

∞ n1

PXn≥an

^∞

n1

PXn≥n^1/t

≤C^∞

n1

P

2Xn^rt≥n^r Xn^rt

< C·^∞

n1

E Xn^rt n^r Xn^rt

<∞ 2.23 by2.21withrpfor 0< t <1 orr2/tfor 1≤t <2.

Let now 0 < t <1. Then, forϕnx x^t,x > 0,n≥ 1, the conditionsAandCwith 1< p≤2 are fulfilled:

A: ∞ i2

b^−p_i E ϕ^p_iXi ϕ^p_i

bi

ϕ^p_iXi i−1

j1

b^p_jE ϕ^p_jXj ϕ^p_j

bj

ϕ^p_jXj

≤ _∞

i1

E Xi^pt i^p Xi^pt

²

<∞, 2.24 by2.21withrp.

C: ∞ n1

E ϕ^2pn

minXn, an ϕ^2pn

bn

ϕ^2pn Xn

^∞

n1

E

minXn, an_2pt n^2p Xn^2pt

≤^∞

n1

E Xn^2pt

n^2p Xn^2ptIXn≥n^1/t ∞

n1

E

Xn^2pt Xn^pt

n^p/Xn^pt

·n^p Xn^ptIXn<n^1/t

≤C^∞

n1

PXn≥n^{1/t ∞}

n1

E Xn^pt n^p Xn^pt

<∞

2.25 by2.23and2.21withr p, 1< p≤2.

Thus, byTheorem 2.1, we have n^−1/tn

i1

Xi−E

XiIXi< i^1/t

−→0 a.s.asn−→ ∞ 2.26

for 0< t <1.

Now, we need to show that2.26also holds for 1≤t <2.

Letϕnx x². Then, forp2, by the similar calculations as in case 0< t <1, we get A₁

: ∞ i2

b^−p_i E ϕiXi ϕi

bi

ϕiXii−1 j1

b^p_jE ϕjXj ϕj

bj

ϕjXj

≤ _∞

i1

E Xi² i^2/t Xi²

²

<∞ 2.27 by2.21withr2/tand

C1 :

∞ n1

E ϕ²_n

minXn, an ϕ²n

bn

ϕ²nXn

^∞

n1

E

minXn, an4

n^4/t Xn⁴

≤C^∞

n1

PXn≥n^{1/t ∞}

n1

E Xn² n^2/t Xn²

<∞ 2.28 by2.23and2.21withr 2/t.

Therefore, byTheorem 2.1, we get2.26for 1≤t <2.

This completes the proof ofCorollary 2.2.

Corollary 2.3. Let{X, Xn, n ≥ 1}be a sequence of ρ^∗-mixing random variables with E|X|^t < ∞, 0< t <2. Let

PXi> x

≤CP

|X|> x

2.29 for allx >0,i≥1 and some positive constantC.

Moreover, when 1≤t <2, letEX0. Then, n^−1/tn

i1

Xi−→0 a.s. asn−→ ∞. 2.30

Proof. We first note that2.29implies EXi^sIXi< a

≤C E

|X|^sI

|X|< a a^sP

|X| ≥a

2.31 for anya >0 ands >0.

Put nowanbnn^1/tfor any 0< t <2.

It is easy to see thatE|X|^t < ∞,2.29and2.31withs rtimply convergence of the series:

∞

n1E Xn^rt

n^r Xn^rt

2.32 forrp,1< p≤2in the case 0< t <1 andr2/tin the case 1≤t <2.

We have ∞

n1

E Xn^rt n^r Xn^rt

^∞

n1

E Xn^rt

n^r Xn^rtIXn< n^1/t _∞

n1

E Xn^rt

n^r Xn^rtIXn≥n^1/t

≤^∞

n1

EXn^rt

n^r IXn< n^{1/t ∞}

n1

PXn≥n^1/t

≤C _∞

n1

E|X|^rt n^r I

|X|< n^{1/t ∞}

n1

P

|X| ≥n^1/t

≤C _∞

n1

E

|X|^{t tr−1}I

|X|< n^1/t

n^{1 1/t·tr−1} E|X|^t

<∞.

2.33 Because ofCorollary 2.2, this proves that2.26holds for any 0< t <2.

To complete the proof we should show that n^−1/t

n i1

E

XiIXi< i^1/t

−→0 asn−→ ∞ 2.34 for any 0< t <2.

For 0< t <1 andI1:n^−1/t_n

i1i^1/tP|X| ≥i^1/t, we have n^−1/t

n

i1

E

XiIXi< i^1/t

≤n^−1/tn

i1

EXiIXi< i^1/t

≤C

n^−1/tn

i1

E

|X|I

|X|< i^1/t I1

≤C

n^−1/tn

i1

E

|X|I

|X|< n^1/t I1

≤C

n^1−1/tE

|X|I

|X|< n^1/t I1

C

n^1−1/tn

j1

E

|X|I

j−1≤ |X|^t< j I1

.

2.35

But

∞ n1

P

|X| ≥n^1/t

≤CE|X|^t<∞, 2.36

so by Kronecker’s lemma we get I1:n^−1/tn

i1

i^1/tP

|X| ≥i^1/t

−→0 asn−→ ∞. 2.37

Moreover, we note ∞

j1

j^1−1/tE

|X|I

j−1≤ |X|^t< j

≤^∞

j1

E

|X|I

j−1≤ |X|^t< j

E|X|^t<∞, 2.38

and by Kronecker’s lemma n^1−1/tn

j1

E

|X|I

j−1≤ |X|^t< j

−→0 asn−→ ∞, 2.39

which together with2.35and2.37gives2.34for 0< t <1.

Let 1≤t <2. First, we will show that

n→∞limP

n i1

Xi

> εn^1/t

0. 2.40

To achieve this, we putYiXiI|Xi|< n^1/tfor 1≤i≤n.

Because ofEXi0 andE|X|^t<∞, we have n^−1/t

n

i1

EYi

≤n^−1/tn

i1

EXiIXi≥n^1/t

≤CE

|X|^tI

|X| ≥n^1/t

−→0 asn−→ ∞.

2.41

Therefore, for large enoughn, we obtain P

n

i1

Xi

> εn^1/t

≤ⁿ

i1

PXi≥n^1/t P

n

i1

Yi−EYi > εn^1/t

. 2.42

Hence, we need only to show that n

i1

PXi≥n^1/t

−→0 asn−→ ∞, 2.43

P

n i1

Yi−EYi > εn^1/t

−→0 asn−→ ∞. 2.44

ByE|X|^t<∞, we get ∞ n1

1 n

n i1

PXi≥n^1/t

≤C^∞

n1

P

|X| ≥n^1/t

≤CE|X|^t<∞, 2.45 which implies2.43.

ByLemma 1.3and2.23withs2, ∞

n1

1 nP

n

i1

Yi−EYi > εn^1/t

≤C^∞

n1

n^{−1 2/tn}

i1

EY_i²

≤C _∞

n1

n^{−1 2/tn}

i1

E X²I

|X|< n^{1/t ∞}

n1

n^{−1 2/tn}

i1

n^2/tP

|X| ≥n^1/t

≤C _∞

n1

n^−2/tn

j1

E X²I

j−1≤ |X|^t< j ^∞

n1

P

|X| ≥n^1/t

≤C _∞

k1

E X²I

k−1≤ |X|^t< k^∞

nk

n^−2/t E|X|^t

≤C _∞

k1

k^{−2/t 1}E X²I

k−1≤ |X|^t < k E|X|^t

≤C _∞

k1

kP

k−1≤ |X|^t< k E|X|^t

< CE|X|^t<∞

2.46

which gives2.44.

Thus, we have established that2.40holds true. Equations2.40and2.26imply2.34 which completes the proof ofCorollary 2.3.

Corollary 2.3gives the Marcinkiewicz SLLN for ρ^∗-mixing random variables {Xn, n ≥ 1}stochastically dominated by a random variable X for 0 < t < 2. The identical result was obtained by Cai 6 as a consequence of complete convergence theorem. Both of them, for 0 < t < 1, are a special case of more general result of Fazekas and T ´om´acs11obtained for stochastically dominated random variables without assuming any kind of dependence.

Acknowledgment

The author thanks the referees for their comments and suggestions which allowed to improve this paper.

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