-Mixing Random Variables

Volume 2010, Article ID 630608,13pages doi:10.1155/2010/630608

Research Article

Complete Convergence for Weighted Sums of ρ

-Mixing Random Variables

Soo Hak Sung

Department of Applied Mathematics, Pai Chai University, Taejon 302-735, South Korea

Correspondence should be addressed to Soo Hak Sung,[email protected] Received 2 August 2009; Accepted 24 March 2010

Academic Editor: Leonid Shaikhet

Copyrightq2010 Soo Hak Sung. 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.

We obtain the complete convergence for weighted sums ofρ^∗-mixing random variables. Our result extends the result of Peligrad and Gut1999on unweighted average to a weighted average under a mild condition of weights. Our result also generalizes and sharpens the result of An and Yuan 2008.

1. Introduction

In many stochastic models, the assumption that random variables are independent is not plausible. So it is of interest to extend the concept of independence to dependence cases. One of these dependence structures isρ^∗-mixing.

Let {Xn, n ≥ 1} be a sequence of random variables defined on a probability space Ω,F, P, and let F^m_n denote the σ-algebra generated by the random variables X_n, X_n1, . . . , X_m.For any S ⊂ N, defineF_S σX_i, i ∈ S.Given two σ-algebras A,B in F,put

ρA,B sup{corrX, Y;X∈L2A, Y ∈L2B}, 1.1

where corrX, Y EXY −EXEY/

varXvarY.Define theρ^∗-mixing coefficients by

ρ^∗_nsup

ρFS,FT;S, T⊂Nwith distS, T≥n

. 1.2

Obviously, 0≤ρ^∗_n1≤ρ^∗_n≤ρ^∗₀1.The sequence{Xn, n≥1}is calledρ^∗-mixingorρ-mixing if there existsk ∈Nsuch thatρ^∗_k <1.Note that if{X_n, n ≥1}is a sequence of independent random variables, thenρ^∗_n0 for alln≥1.

A number of limit results for ρ^∗-mixing sequences of random variables have been established by many authors. We refer to Bradley1 for the central limit theorem, Bryc and Smole ´nski2 , Peligrad and Gut3 , and Utev and Peligrad4 for moment inequalities, Gan 5 , Kuczmaszewska6 , and Wu and Jiang7 for almost sure convergence, and An and Yuan 8 , Cai9 , Gan5 , Kuczmaszewska10 , Peligrad and Gut3 , and Zhu11 for complete convergence.

The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins12 . A sequence{X_n, n≥1}of random variables converges completely to the constantθif

∞ n1

P|Xn−θ|> <∞ ∀ >0. 1.3

In view of the Borel-Cantelli lemma, this implies thatX_n → θalmost surely. Therefore, the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables as well as weighted sums of random variables. Hsu and Robbins12 proved that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite. Erd ¨os13 proved the converse. The result of Hsu-Robbins-Erd ¨os is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. One of the most important generalizations is Baum and Katz14 strong law of large numbers.

Theorem 1.1 Baum and Katz 14 . Let p ≥ 1/α and 1/2 < α ≤ 1. Let {X_n, n ≥ 1} be a sequence of independent and identically distributed random variables with EX1 0. Then the following statements are equivalent:

iE|X1|^p<∞;

ii_∞

n1n^pα−2Pmax1≤j≤n|j

i1Xi|> n^α<∞for all >0,

Peligrad and Gut [3] extended the result of Baum and Katz [14] toρ^∗-mixing random variables.

Theorem 1.2Peligrad and Gut3 . Letp >1/αand 1/2< α≤1.Let{Xn, n≥1}be a sequence of identically distributedρ^∗-mixing random variables withEX1 0.Then the following statements are equivalent:

iE|X1|^p<∞;

ii_∞

n1n^pα−2Pmax_1≤j≤n|j

i1X_i|> n^α<∞for all >0.

Cai [9] complementedTheorem 1.2whenp1/α.

Recently, An and Yuan [8] obtained a complete convergence result for weighted sums of identically distributedρ^∗-mixing random variables.

Theorem 1.3An and Yuan8 . Letp >1/αand 1/2 < α≤1.Let{Xn, n≥1}be a sequence of identically distributedρ^∗-mixing random variables withEX10.Assume that{a_ni,1≤i≤n, n≥1}

is an array of real numbers satisfying

n i1

|ani|^pO

n^δ for some 0< δ <1, 1.4

#A_nk#

1≤i≤n:|a_ni|^p>k1⁻¹

≥ne^−1/k ∀k≥1, n≥1. 1.5

Then the following statements are equivalent:

iE|X1|^p<∞;

ii_∞

n1n^pα−2Pmax_1≤j≤n|_j

i1a_niX_i|> n^α<∞for all >0.

Note that the result of An and Yuan8 is not an extension of Peligrad and Gut’s3 result, since condition1.4does not hold for the array witha_ni 1,1 ≤ i ≤ n, n ≥ 1.An and Yuan8 proved the implicationi⇒iiunder condition1.4, and proved the converse under conditions1.4and1.5. However, the array satisfying both1.4and1.5does not exist. Noting that #Ank/k1 ≤ _n

i1|ani|^p ≤ On^δ,we have thatne^−1/k ≤ #Ank ≤ k 1On^δ.But, this does not hold whenkis fixed andnis large enough.

In this paper, we obtain a new complete convergence result for weighted sums of identically distributedρ^∗-mixing random variables. Our result extends the result of Peligrad and Gut3 , and generalizes and sharpens the result of An and Yuan8 .

Throughout this paper, the symbol C denotes a positive constant which is not necessarily the same one in each appearance,x denotes the integer part ofx,anda∧b min{a, b}.

2. Main Result

To prove our main result, we need the following lemma which is a Rosenthal-type inequality forρ^∗-mixing random variables.

Lemma 2.1Utev and Peligrad4 . Let{X_n, n≥1}be a sequence ofρ^∗-mixing random variables with EXn 0 andE|Xn|^r < ∞for somer ≥ 2 and all n ≥ 1.Then there exists a constant D Dr, k, ρ_k^∗depending only onr, k,andρ^∗_ksuch that for anyn≥1,

E

⎛

⎝max

1≤j≤n

j i1

Xi

r⎞

⎠≤D

⎧⎨

⎩ n

i1

E|Xi|^r _n

i1

EX_i² _r/2⎫

⎬

⎭, 2.1

whereρ^∗_k<1.

Now we state the main result of this paper.

Theorem 2.2. Letp >1/αand 1/2< α≤1.Let{Xn, n≥1}be a sequence of identically distributed ρ^∗-mixing random variables withEX1 0.Assume that{a_ni,1 ≤ i≤ n, n≥ 1}is an array of real numbers satisfying

n i1

|ani|^qOn for some q > p. 2.2

IfE|X1|^p<∞,then

∞ n1

n^pα−2P

max1≤j≤n

j i1

a_niX_i > n^α

<∞ ∀ >0. 2.3

Conversely, if2.3holds for any array{ani}satisfying2.2, thenE|X1|^p<∞.

To proveTheorem 2.2, we first prove the following lemma which is the sufficiency of Theorem 2.2when the array is bounded.

Lemma 2.3. Let{Xn, n≥1}be a sequence of identically distributedρ^∗-mixing random variables with EX1 0 andE|X1|^p<∞for somep >1/αand 1/2< α≤1.Assume that{a_ni,1≤i≤n, n≥1}is an array of real numbers satisfying|ani| ≤1 for 1≤i≤nandn≥1.Then2.3holds.

Proof. For 1≤i≤nandn≥1,defineX_ni XiI|Xi| ≤n^α.SinceEXi0 and_n

i1|ani| ≤n,we have that

n^−αmax

1≤j≤n

j i1

a_niEX_ni

n^−αmax

1≤j≤n

j i1

a_niEX_iI|X_i|> n^α

≤n^−αn

i1

|a_ni|E|X1|I|X1|> n^α

≤n^1−αE|X1|I|X1|> n^α

≤n^1−pαE|X1|^pI|X1|> n^α−→0

2.4

asn → ∞.Hence fornlarge enough, we have

n^−αmax

1≤j≤n

j i1

aniEX_ni <

2 . 2.5

It follows that

∞ n1

n^pα−2P

max1≤j≤n

j i1

aniXi

> n^α

≤^∞

n1

n^pα−2n

i1

P|X_i|> n^{α ∞}

n1

n^pα−2P

max1≤j≤n

j i1

a_niX_ni > n^α

≤^∞

n1

n^pα−1P|X1|> n^α C^∞

n1

n^pα−2P

max1≤j≤n

j i1

a_ni

X_ni −EX_ni > n^α

2

:ICJ.

2.6

Noting that_∞

n1n^pα−1P|X1|> n^α≤CE|X1|^p<∞,we haveI <∞.Thus, it remains to show thatJ <∞.

We have by Markov’s inequality andLemma 2.1that for anyr ≥2,

J≤ 2

_r∞

n1

n^{pα−rα−2}Emax

1≤j≤n

j i1

ani

X_ni −EX_ni

r

≤C^∞

n1

n^{pα−rα−2}

⎧⎨

⎩ _n

i1

a²_niEX_ni ²_r/2 ⁿ

i1

|a_ni|^rEX_ni^r⎫

⎬

⎭

≤C^∞

n1

n^{pα−rα−2r/2}

E|X1|²I|X1| ≤n^{α r/2}C^∞

n1

n^{pα−rα−1}E|X1|^rI|X1| ≤n^α :CJ1CJ2.

2.7

In the last inequality, we used the fact that|a_ni| ≤1 for 1≤i≤nandn≥1.

Ifp ≥2,then we take large enoughr such thatr > max{pα−1/α−1/2, p}.Since r >pα−1/α−1/2,we get

J1≤C^∞

n1

n^{pα−rα−2r/2}<∞. 2.8

Sincer > p,we also get

J2^∞

n1

n^{pα−rα−1n}

i1

E|X1|^rI

i−1^α<|X1| ≤i^α

^∞

i1

E|X1|^rI

i−1^α<|X1| ≤i^α∞

ni

n^{pα−rα−1}

≤C^∞

i1

E|X1|^rI

i−1^α<|X1| ≤i^α i^pα−rα

≤CE|X1|^p<∞.

2.9

Ifp <2,then we taker 2.Sincer > p,2.9still holds, and soJ1J2<∞.

We next prove the sufficiency ofTheorem 2.2when the array is unbounded.

Lemma 2.4. Let{X_n, n≥1}be a sequence of identically distributedρ^∗-mixing random variables with EX1 0 andE|X1|^p<∞for somep >1/αand 1/2< α≤1.Assume that{a_ni,1≤i≤n, n≥1}is an array of real numbers satisfyingani0 or|ani|>1,and

n i1

|a_ni|^q≤n for some q > p. 2.10

Then2.3holds.

Proof. Ifp <2,then we can takeδ >0 such thatp < pδ <min{2, q}.Sinceani0 or|ani|>1, we have that_n

i1|a_ni|^pδ ≤ _n

i1|a_ni|^q ≤n.Thus we may assume that2.10holds for some p < q <2 whenp <2.

LetS_njj

i1a_niX_iI|a_niX_i| ≤n^αfor 1≤j≤nandn≥1.In view ofEX_i0,we get

n^−αmax

1≤j≤n

ES_njn^−αmax

1≤j≤n

j i1

aniEXiI|aniXi|> n^α

≤n^−αn

i1

E|a_niX_i|I|a_niX_i|> n^α

≤n^−pαn

i1

E|a_niX_i|^pI|a_niX_i|> n^α

≤n^−pαn

i1

|a_ni|^pE|X1|^p

≤n^−pα _n

i1

|a_ni|^q _p/q

n^1−p/qE|X1|^p

≤n^1−pαE|X1|^p−→0,

2.11

sincepα >1.Hence fornlarge enough, we have thatn^−αmax_1≤j≤n|ES_nj|< /2.It follows that

∞ n1

n^pα−2P

max1≤j≤n

j i1

a_niX_i > n^α

≤^∞

n1

n^pα−2P

max1≤i≤n|aniXi|> n^α

^∞

n1

n^pα−2P

max1≤j≤n

S_nj> n^α

≤^∞

n1

n^pα−2n

i1

P|a_niX_i|> n^α C^∞

n1

n^pα−2P

max1≤j≤n

S_nj−ES_nj> n^α 2

:ICJ.

2.12

For 1≤j≤n−1 andn≥2,let

I_nj

1≤i≤n:n^1/q

j1_−1/q

<|a_ni| ≤n^1/qj^−1/q

. 2.13

Then{Inj,1≤j ≤n−1}are disjoint,_n−1

j1Inj {1≤i≤n:ani/0},and_k

j1#Inj ≤k1 for 1≤k≤n−1,since

n≥

{1≤i≤n:ani/0}

|ani|^qn−1

j1

i∈Inj

|ani|^q≥n^k

j1

1

j1#Inj ≥ n k1

k j1

#Inj. 2.14

For convenience of notation, lett1/α−1/q.Sinceani 0 or|ani|>1,and_n

i1|ani|^q ≤n, we havea11 0.It follows that

I^∞

n2

n^pα−2n−1

j1

i∈Inj

P|aniXi|> n^α

≤^∞

n2

n^pα−2n−1

j1

P

|X1|^t> nj^t/q #I_nj

≤^∞

n2

n^pα−2n−1

j1

#I_nj

k≥nj^t/q

P

k <|X1|^t≤k1

≤^∞

n2

n^pα−2∞

kn

P

k <|X1|^t≤k1

n−1∧k1/n ^q/t

j1

#I_nj

≤^∞

n2

n^pα−2∞

kn

P

k <|X1|^t≤k1 n∧k1 n

_q/t 1

≤^∞

n1

n^pα−2n 1t/q

kn

P

k <|X1|^t≤k1 k1 n

_q/t 1

^∞

n1

n^{pα−1 ∞}

kn^1t/q 1

P

k <|X1|^t≤k1

:I1I2.

2.15

Sincepα−2−q/t−αq−p−1<−1,we obtain

I1≤C^∞

n1

n^{pα−2−q/tn} 1t/q

kn

P

k <|X1|^t≤k1 k^q/t

≤C^∞

k1

P

k <|X1|^t≤k1 k^{q/t k}

nk^q/qt

n^{pα−2−q/t}

≤C^∞

k1

P

k <|X1|^t≤k1 kq/t−qαq−p/qt

≤CE|X1|^p<∞.

2.16

We also obtain

I2≤^∞

k1

P

k <|X1|^t≤k1

k^q/tq

n1

n^pα−1

≤C^∞

k1

P

k <|X1|^t≤k1 k^pα−1/q≤CE|X1|^p<∞.

2.17

FromI1<∞andI2 <∞,we haveI <∞.Thus, it remains to show thatJ <∞.

We have by Markov’s inequality andLemma 2.1that for anyr ≥2,

J≤C^∞

n1

n^{pα−rα−2}Emax

1≤j≤n

S_nj−ES_nj^r

≤C^∞

n1

n^{pα−rα−2} _n

i1

E|aniXi|²I|aniXi| ≤n^α _r/2

C^∞

n1

n^{pα−rα−2n}

i1

E|aniXi|^rI|aniXi| ≤n^α :J1J2.

2.18

Observe that forr≥qandn > m,

n≥ⁿ⁻¹

j1

i∈Inj

|ani|^q≥nⁿ⁻¹

j1

1

j1#Inj≥nm1^r/q−1n−1

jm

j1_−r/q

#Inj. 2.19

So_n−1

jmj^−r/q#Inj≤Cm^−r/q−1forr ≥qandn > m.

ForJ1andJ2,we proceed with two cases.

iIfp≥2,then we takerlarge enough such thatr >max{pα−1/α−1/2, q}.Then we obtain that

J1≤C^∞

n1

n^{pα−rα−2} _n

i1

|a_ni|² _r/2

≤C^∞

n1

n^{pα−rα−2} _n

i1

|ani|^q _r/2

≤C^∞

n1

n^{pα−rα−2r/2}<∞.

2.20

The second inequality follows by the fact thata_ni 0 or|a_ni|>1.

Noting thata110,we also obtain that

J2^∞

n2

n^{pα−rα−2n−1}

j1

i∈Inj

E|aniXi|^rI|aniXi| ≤n^α

≤^∞

n2

n^{pα−rα−2r/qn−1}

j1

j^−r/q#InjE|X1|^rI

|X1|^t≤n

j1_t/q

≤^∞

n2

n^{pα−rα−2r/qn−1}

j1

j^−r/q#I_nj

0≤k≤nj1^t/q

E|X1|^rI

k <|X1|^t≤k1

^∞

n2

n^{pα−rα−2r/qn−1}

j1

j^−r/q#I_nj²ⁿ

k0

E|X1|^rI

k <|X1|^t≤k1

^∞

n2

n^{pα−rα−2r/qn−1}

j1

j^−r/q#I_nj^nj1 t/q

k2n1

E|X1|^rI

k <|X1|^t≤k1

:J3J4.

2.21

Sincepα−rα−2r/q < qα−rα−2r/q−r−qα−1/q−1<−1 andq > p,we have that

J3 ^∞

n2

n^{pα−rα−2r/q2n}

k0

E|X1|^rI

k <|X1|^t≤k1

n−1

j1

j^−r/q#Inj

≤C^∞

n2

n^{pα−rα−2r/q2n}

k0

E|X1|^rI

k <|X1|^t≤k1

≤C^∞

k1

E|X1|^rI

k <|X1|^t≤k1 ∞ nk/2

n^{pα−rα−2r/q}

≤C^∞

k1

E|X1|^rI

k <|X1|^t≤k1 k^{pα−rα−1r/q}

≤C^∞

k1

P

k <|X1|^t≤k1 k^pα−1

≤CE|X1|^p<∞.

2.22

Since 1/t1/q−α0 andpα−2−q/t−αq−p−1<−1,we also have that

J4 ≤^∞

n2

n^{pα−rα−2r/qn}

qt/q

k2n1

E|X1|^rI

k <|X1|^t≤k1

n−1

jk/n^q/t −1

j^−r/q#I_nj

≤C^∞

n2

n^{pα−rα−2r/qn}

qt/q

k2n1

E|X1|^rI

k <|X1|^t≤k1 k n

_q/t

−1

_−r/q−1

≤C^∞

k5

E|X1|^rI

k <|X1|^t≤k1 k^{−r−q/t k/2}

nk^q/qt

n^{pα−2−q/t}

≤C^∞

k5

E|X1|^rI

k <|X1|^t≤k1 k−r−q/t−α−1/qq−p

≤CE|X1|^p<∞.

2.23

FromJ3<∞andJ4<∞,we haveJ2<∞.

iiIfp <2,then we taker 2.As noted above, we may assume thatp < q <2.Since r > q,as in the casep≥2,we haveJ1J2≤CE|X1|^p<∞.

We now proveTheorem 2.2by using Lemmas2.3and2.4.

Proof ofTheorem 2.2.

Sufficiency. Without loss of generality, we may assume that_n

i1|ani|^q ≤nfor someq > p.For n≥1,let

An{1≤i≤n:|ani| ≤1}, Bn{1≤i≤n:|ani|>1}, 2.24

and leta_nianiifi∈An, a_ni0 otherwise, anda_nianiifi∈Bn, a_ni0 otherwise. Then

max1≤j≤n

j i1

a_niX_i ≤max

1≤j≤n

j i1

a_niX_i max

1≤j≤n

j i1

a_niX_i

. 2.25

It follows that ∞ n1

n^pα−2P

max1≤j≤n

j i1

aniXi

> n^α

≤^∞

n1

n^pα−2P

max1≤j≤n

j i1

a_niXi

> n^α

2

^∞

n1

n^pα−2P

max1≤j≤n

j i1

a_niXi

> n^α

2

:IJ.

2.26

ByLemma 2.3, we haveI <∞.ByLemma 2.4, we haveJ <∞.Hence2.3holds.

Necessity. Choose, for eachn≥ 1, an1 · · · ann 1.Then{ani}satisfies2.2. By2.3, we obtain that

∞ n1

n^pα−2P

max1≤j≤n

j i1

X_i > n^α

<∞ ∀ >0, 2.27

which implies that

∞ n1

n^pα−2P

max1≤j≤nXj> n^α

<∞ ∀ >0. 2.28

Observe that

∞>^∞

i1 2ⁱ

n2ⁱ⁻¹1

n^pα−2P

max1≤j≤nXj> n^α

≥

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ ∞

i1

2ⁱ⁻¹_pα−2 2ⁱ⁻¹P

1≤j≤2maxⁱ⁻¹Xj>

2ⁱ_α

ifpα≥2, ∞

i1

2ⁱ_pα−2 2ⁱ⁻¹P

1≤j≤2maxⁱ⁻¹Xj>

2ⁱ_α

if 1< pα <2,

≥

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ ∞

i1

P

1≤j≤2maxⁱ⁻¹Xj>

2ⁱ_α

ifpα≥2,

2^pα−2 ∞

i1

P

1≤j≤2maxⁱ⁻¹Xj>

2ⁱ_α

if 1< pα <2.

2.29

Hence we have that for any > 0, Pmax_1≤j≤2ⁱ⁻¹|X_j| > 2^iα → 0 as i → ∞, and so Pmax1≤j≤n|Xj| > n^α → 0 as n → ∞. The rest of the proof is same as that of Peligrad and Gut3 and is omitted.

Remark 2.5. Takingani 1 for 1 ≤ i ≤ nandn ≥ 1,we can immediately getTheorem 1.2 fromTheorem 2.2. If the array{ani}satisfies1.4, then it satisfies 2.2: takingqsuch that p < q < p/δ,we have

n i1

|a_ni|^q≤max

1≤i≤n|a_ni|^q−pn

i1

|a_ni|^p≤Cn^δq−p/pn^δ≤Cn. 2.30

So the implication i⇒ii of Theorem 1.3 follows from Theorem 2.2. As noted after Theorem 1.3, the implicationii⇒iofTheorem 1.3is not true. Therefore, our result extends the result of Peligrad and Gut3 to a weighted average, and generalizes and sharpens the result of An and Yuan8 .

Acknowledgments

The author is grateful to the editor Leonid Shaikhet and the referees for the helpful comments and suggestions that considerably improved the presentation of this paper. This work was supported by the Korea Science and Engineering FoundationKOSEFGrant funded by the Korea governmentMOST no. R01-2007-000-20053-0.

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