Strong Convergence Properties for Asymptotically Almost Negatively Associated Sequence

(1)

Discrete Dynamics in Nature and Society Volume 2012, Article ID 562838,8pages doi:10.1155/2012/562838

Research Article

Strong Convergence Properties for Asymptotically Almost Negatively Associated Sequence

Xueping Hu,

^{1, 2}

Guohua Fang,

²

and Dongjin Zhu

³

1School of Mathematics and Computational Science, Anqing Teachers College, Anqing 246133, China

2College of Water Conservancy and Hydropower Engineering, HoHai University, Nanjing 210098, China

3College of Mathematics and Computation Science, Anhui Normal University, Wuhu 241000, China

Correspondence should be addressed to Xueping Hu,[email protected] Received 22 June 2012; Accepted 10 September 2012

Academic Editor: Garyfalos Papaschinopoulos

Copyrightq2012 Xueping Hu et al. 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.

By applying the moment inequality for asymptotically almost negatively associatedin short AANA random sequence and truncated method, we get the three series theorems for AANA random variables. Moreover, a strong convergence property for the partial sums of AANA random sequence is obtained. In addition, we also study strong convergence property for weighted sums of AANA random sequence.

1. Introduction

A finite family of random variables{Xk,1 ≤k≤n, n≥2}is said to be negatively associated in short NAif for every pair of disjoint subsetsA₁, A₂of{1,2, . . . , n}

Cov

fXi:i∈A1, g

Xj:j∈A2

≤0, 1.1

wheneverf, gare coordinate-wise nondecreasing such that the covariance exists. An infinite sequence of random variables{Xn, n≥1}is said to be NA if every finite subfamily is NA.

The notion of NA was first introduced by Block et al.1982 1. Joag-Dev and Proschan 1983 2showed that many well-known multivariate distributions possess the NA property.

By inspecting the proof of maximal inequality for NA random variables in Matuła 3, Chandra and Ghosal discovered that one can also allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal4,5introduced the following dependence.

(2)

Definition 1.1. A sequence{Xn, n≥1}of random variables is said to be asymptotically almost negatively associated, if there exists a nonnegative sequenceqn → 0 asn → ∞such that

Cov

fXn, gXn1, X_n2, . . . , X_nk

≤qn

VarfXnVargXn1, X_n2, . . . , X_nk_1/2 ,

1.2 for all n, k ≥ 1 and for all coordinatewise nondecreasing continuous functions f and g whenever the variances exit.

Obviously, the family of AANA sequences contain NAin particular, independent sequenceswithqn 0, n≥ 1and some more sequences of random variables which are not much deviated from being NA. An example of an AANA sequence which is not NA was introduced by Chandra and Ghosal4.

Since the notion of AANA sequence was introduced by Chandra and Ghosal4, the AANA properties have aroused wide interest because of numerous applications in reliability theory, percolation theory, and multivariate statistical analysis. In the past decades, a lot of eﬀort was dedicated to proving the limit theorems of AANA random variables; we can refer to 4–10. Hence, extending the limit properties of AANA random variables has very important significance in the theory and application.

In this paper, we mainly study the strong convergence property for the partial sums of AANA random variables; furthermore the strong convergence property for weighted sums of AANA random variables is also obtained.

Throughout the paper, letIA be the indicator function of the setA, and let X^c

−cIX < −c XI|X| ≤ c cIX > cfor somec > 0. Thean Obndenotes that there exits a positive constantCsuch that|an/b_n| ≤C. The symbolCrepresents a positive constant which may be diﬀerent in various places. The main results of this paper are dependent on the following lemmas.

Lemma 1.2 Yuan and An 6. Let {Xn, n ≥ 1} be a sequence of AANA random variables with mixing coeﬃcients{qn, n ≥ 1}, and let f₁, f₂, . . . be all nondecreasing (or nonincreasing) functions; then{fnXn, n≥1}is still a sequence of AANA random variables with mixing coeﬃcients {qn, n≥1}.

Lemma 1.3Wang et al.7. For 1 < p ≤ 2, let {Xn, n ≥ 1}be a sequence of AANA random variables with mixing coeﬃcients{qn, n≥1}andEXn0 for eachn≥1. If_∞

n1q²n<∞, then there exists a positive constantC_pdepending only onpsuch that

E

max1≤i≤n|Si|^p

≤C_p

n i1

E|Xi|^p, 1.3

for alln≥1 whereS_i _i

j1X_j, C_p2^p2^2−pp 6p^p_∞

n1q²n^p/q, andqp/p−1is the dual number ofp.

Lemma 1.4Wu11. Let{Xn, n ≥ 1}be a sequence of random variables. For eachn ≥ 1, there exists a random variableXsuch that

P|Xn| ≥x≤CP|X| ≥x 1.4

(3)

then, for anyr >0, x >0, the following two statements hold:

E|Xn|^rI|Xn| ≤x≤C

E|X|^rI|X| ≤x x^rP|X|> x , E|Xn|^rI|Xn|> x≤C

E|X|^rI|X|> x

. 1.5

Lemma 1.5Sung12. Letφxbe a positive increasing function on0,∞satisfyingφx↑ ∞ asn → ∞, and letψxbe the inverse function ofφx. Ifψxandφxsatisfy, respectively,

ψn ⁿ

i1

1

ψiOn, E

φ|X|

<∞, 1.6

then

∞ i1

1

ψnE|X|I

|X|> ψn

<∞. 1.7

2. Strong Convergence for the Partial Sums of AANA Random Variables

Theorem 2.1. Let{Xn, n ≥ 1}be a sequence of AANA random variables with_∞

n1q²n< ∞, if the following assumptions holds:

∞ n1

P|Xn|> c<∞, ^∞

n1

EX_n^c<∞, ^∞

n1

VarX_n^c<∞; 2.1

then_∞

n1X_nalmost surely convergence.

Remark 2.2. The proof ofTheorem 2.1is similar to the proof of Theorem 4.3.4 in11, and by Lemmas1.2and1.3, we omit it.

Theorem 2.3. Let{Xn, n≥1}be a sequence of AANA random variables with_∞

n1q²n<∞.

Assume that{gnx, n≥1}is a sequence of even functions inR¹, for eachn≥1,gnxis a positive nondecreasing function in0,∞and satisfies one of the following conditions:

iforx∈0,1there exists a constantα >0 such thatgnx≥αx;

iiforx∈0,1, there exists a constantr∈1,2andα >0 such thatgnx≥αx^r; however, forx∈1,∞, gnx≥αx, furthermore assume thatEX_n0, for eachn≥1.

Let{an, n≥1}be a constant sequence satisfying 0< an↑ ∞such that

∞ n1

Egn

Xn

a_n

<∞, 2.2

(4)

then_∞

n1Xn/analmost surely convergence, and further it follows from the “Kronecker lemma”

that

a⁻¹_n

n k1

Xk−→0a.s., as n−→ ∞. 2.3

Proof. For eachn≥1, denoteX_n^aⁿ −anIXn<−an X_nI|Xn| ≤a_n a_nIXn> a_n.

By Lemma 1.2, we can see that, for fixed n ≥ 1, {Xn^aⁿ} is still a sequence of AANA random variables. To verity theTheorem 2.3, forc1 we only need to prove the convergence of three series of 2.1 under condition i or ii. The proof of Theorem 2.3 includes the following three steps.

1We prove_∞

n1P|Xn/an|>1<∞under condition (i) or (ii).

For each n ≥ 1, if g_nx satisfies condition i, noting that g_nx is a positive nondecreasing even function in0,∞, it is obvious that

P

X_n an

>1

EI X_n

an

>1

≤α⁻¹Eg_n X_n

an

. 2.4

By2.2, we can get

∞ n1

P

X_n an

>1

≤α⁻¹

∞ n1

Egn

X_n an

<∞. 2.5

Ifgnxsatisfies conditionii, it is easy to prove that2.5also holds when|Xn|> an>0.

2Next we will show_∞

n1E|X_n^aⁿ/an|<∞.

Ifg_nxsatisfies conditioni, it follows that

EX^a_nⁿ a_n

−EIXn<−an EXn

a_nI|Xn| ≤a_n EIXn> a_n

≤EI|Xn|> an

EX_n

anI|Xn| ≤an

≤α⁻¹Eg_n X_n

an

|Xn|≤an

X_n an

dP

≤2α⁻¹Eg_n X_n

a_n

.

2.6

(5)

On the other hand, if conditioniiholds, according toEXn0, for eachn≥1, we have EX_n^aⁿ

a_n

≤EI|Xn|> an

EXn

a_nI|Xn| ≤an EI|Xn|> a_n

EX_n

anI|Xn|> a_n

≤2α⁻¹Egn

Xn

a_n

.

2.7

Hence, it follows from2.2that

∞ n1

E X^a_nⁿ

an

<2α⁻¹

∞ n1

Eg_n X_n

an

<∞. 2.8

3Finally we prove_∞

n1EX^anⁿ/a_n²<∞.

If gnx satisfies condition i, for each n ≥ 1, it is easy to show that by the C_r−inequality

E X^a_nⁿ

an

2

E

−IXn<−an

X_n

anI|Xn| ≤an IXn> an ²

≤3E

I|Xn|> a_n X_n

an

₂

I|Xn| ≤a_n

≤Cα⁻¹Eg_n X_n

a_n

CE X_n

a_n

I|Xn| ≤a_n

≤Cα⁻¹Egn

Xn

a_n

.

2.9

If conditioniiholds, according to theC_r-inequality, for eachn≥1, we get

E X_n^aⁿ

a_n 2

E

−IXn<−an

X_n

a_nI|Xn| ≤a_n IXn> a_n ²

≤3E

I|Xn|> an

Xn

a_n 2

I|Xn| ≤an

≤Cα⁻¹Egn

X_n an

CE

X_n an

^rI|Xn| ≤an

≤Cα⁻¹Eg_n X_n

a_n

.

2.10

(6)

Therefore, it also follows from2.2that

∞ n1

E X^anⁿ

an

2

< Cα⁻¹

∞ n1

Egn

Xn

an

<∞. 2.11

The proof of theTheorem 2.3is completed by2.5,2.8, and2.11.

Corollary 2.4. Let{Xn, n ≥ 1}be a sequence of AANA random variables with _∞

n1q²n < ∞, and let {an, n ≥ 1} be a constant sequence satisfying 0 < a_n ↑ ∞. For θ ∈ 0,1, let g_nx

|x|^θ/1|x|^θ, and if{Xn/an, n≥1}satisfies2.2, thena⁻¹_n _n

k1Xk → 0 a.s., asn → ∞.

Proof. It is easy to check that{gnx, n ≥ 1}is a sequence of even functions inR¹, for each n≥1,g_nxis a positive nondecreasing function in0,∞, and the following condition holds:

g_nx≥ 1 2x^θ≥ 1

2x, 0< x≤1, 0< θ≤1. 2.12

3. Strong Convergence for the Weighted Sums of AANA Random Variables

Theorem 3.1. Let{Xn, n≥1}be a diﬀerent distribution sequence of AANA random variables with _∞

n1q²n<∞andEXn 0, for eachn≥1. There exists a random variableXsatisfyingE|X|^r <

∞,0< r≤2, such that

P|Xn|> x≤CP|X|> x, n≥1, x >0. 3.1

Assume that the following conditions hold for the constant arrays{ani, n≥1, 1≤i≤n}.

(i) max_1≤i≤n|ani| Oψ⁻¹n; (ii) for some constantδ > 0,_n

i1|ani|^r On⁻¹log^−1−δn, whereφx, ψxsatisfyLemma 1.5; then

T_n ⁿ

i1

a_niX_i−−−→^a.s. 0, n−→ ∞. 3.2

Proof. LetY_i−ψnIXi<−ψn X_iI|Xi| ≤ψn ψnIXi > ψn, YiY_i−EY_i:

Tn ⁿ

i1

aniXi−Yi n i1

aniYi ⁿ

i1

aniEYiTn1Tn2Tn3. 3.3 It suﬃces to prove thatTni → 0 a.s., as n → ∞, i 1,2,3. We will estimate each of these terms separately.

(7)

To verityTn1 → 0 a.s., asn → ∞, we can get from3.1andEφ|X|<∞that

∞ n1

PXi/Yi

∞ n1

P

|Xi|> ψn

≤C

∞ n1

P

|X|> ψn C

∞ n1

P

φ|X|> n

≤CEφ|X|<∞.

3.4

Hence, by the Borel-Cantelli Lemma it is obvious thatTn1 → 0 a.s., asn → ∞.

Next we will show thatTn2 → 0 asn → ∞almost surely. For anyε > 0,0 < r ≤ 2, note thatE|X|^r < ∞, and it follows from the Markov inequality,Lemma 1.2,Lemma 1.3, Cr-inequality, andLemma 1.5that

∞ n1

P _n

i1

aniYi> ε

≤C

∞ n1

E

n i1

aniYi

r

≤C

∞ n1

n i1

EaniYi^r

≤C

∞ n1

n i1

|ani|^r

E|Xi|^rI

|Xi| ≤ψn

ψ^rnEI

|Xi|> ψn

≤C

∞ n1

n i1

|ani|^r E|X|^rI

|X| ≤ψn

ψ^rnEI

|X|> ψn

≤C

∞ n1

n i1

|ani|^r EX|^rI

|X| ≤ψn

E|X|^r

≤C

∞ n1

n i1

|ani|^r

≤C

∞ n1

1

nlog^1δn <∞.

3.5

the last series converges using conditionii, and by Borel-Cantelli lemma we getT_n2 → 0 a.s., asn → ∞.

Finally we will prove thatTn3 → 0 a.s., asn → ∞. Note thatEXn0; for eachn≥1, it is easy to show that byLemma 1.5,Lemma 1.4, and theKronecker lemma

n i1

Ea_niY_i ≤

n i1

Ea_niX_iI

|Xi| ≤ψn

n i1

a_niψnEI

|Xi|> ψn

≤

n i1

Ea_niX_iI

|Xi|> ψn

n i1

a_niψnEI

|Xi|> ψn

≤C

n i1

E|aniXi|I

|Xi|> ψi

(8)

≤C

n i1

E|aniX|I

|X|> ψi

≤ 1 ψn

n i1

E|X|I

|X|> ψi

−→0, n−→ ∞.

3.6 The proof ofTheorem 3.1is completed.

Acknowledgments

This paper is supported by the National Natural Science Foundantion of China10901003 and the Natural Science Foundation of Anhui Province KJ2012ZD001, KJ2013A126, KJ2012Z233.

References

1 H. W. Block, T. H. Savits, and M. Shaked, “Some concepts of negative dependence,” The Annals of Probability, vol. 10, no. 3, pp. 765–772, 1982.

2 K. Joag-Dev and F. Proschan, “Negative association of random variables, with applications,” The Annals of Statistics, vol. 11, no. 1, pp. 286–295, 1983.

3 P. Matuła, “A note on the almost sure convergence of sums of negatively dependent random variables,” Statistics & Probability Letters, vol. 15, no. 3, pp. 209–213, 1992.

4 T. K. Chandra and S. Ghosal, “The strong law of large numbers for weighted averages under dependence assumptions,” Journal of Theoretical Probability, vol. 9, no. 3, pp. 797–809, 1996.

5 T. K. Chandra and S. Ghosal, “Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables,” Acta Mathematica Hungarica, vol. 71, no. 4, pp. 327–336, 1996.

6 D. Yuan and J. An, “Rosenthal type inequalities for asymptotically almost negatively associated random variables and applications,” Science in China A, vol. 52, no. 9, pp. 1887–1904, 2009.

7 X. Wang, S. Hu, and W. Yang, “Convergence properties for asymptotically almost negatively associated sequence,” Discrete Dynamics in Nature and Society, vol. 2010, Article ID 218380, 15 pages, 2010.

8 X. Wang, S. Hu, and W. Yang, “Complete convergence for arrays of rowwise asymptotically almost negatively associated random variables,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 717126, 11 pages, 2011.

9 Y. Wang, J. Yan, F. Cheng, and C. Su, “The strong law of large numbers and the law of the iterated logarithm for product sums of NA and AANA random variables,” Southeast Asian Bulletin of Mathematics, vol. 27, no. 2, pp. 369–384, 2003.

10 J. Baek II, “Almost sure convergence for asymptotically almost negatively associated random variables sequence,” Communications of the Korean Statistical Society, vol. 16, no. 6, pp. 1013–1022, 2009.

11 Q. Y. Wu, “Probability limit theory for mixing sequence,” Sciences Press, 2005Chinese.

12 S. H. Sung, “Strong laws for weighted sums of i.i.d. random variables. II,” Bulletin of the Korean Mathematical Society, vol. 39, no. 4, pp. 607–615, 2002.

(9)

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

^{Journal of}

in Engineering

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Probability and Statistics

Journal of

Mathematical PhysicsAdvances in

Complex Analysis

^{Journal of}

Optimization

^{Journal of}

Combinatorics

Operations Research

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

The Scientific World Journal

Algebra

Discrete Dynamics in Nature and Society

Decision Sciences

Discrete Mathematics

^{Journal of}

Strong Convergence Properties for Asymptotically Almost Negatively Associated Sequence

Research Article