A note on the rates of convergence for weighted sums of ρ * -mixing random variables

Lithuanian Mathematical Journal, Vol. 54, No. 2, April, 2014, pp. 220–228

A note on the rates of convergence for weighted sums

of ρ∗-mixing random variables

Yongfeng Wu a,1,2, Soo Hak Sung b, and Andrei Volodin c

a College of Mathematics and Computer Science, Tongling University, Tongling 244000, PR Chinab Department of Applied Mathematics, Pai Chai University, Taejon 302735, South Korea

c Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan S4S 0A2, Canada(e-mail: [email protected]; [email protected]; [email protected])

Received May 14, 2013; revised December 23, 2013

Abstract. We discuss the rates of convergence for weighted sums of ρ∗-mixing random variables. We solve an openproblem posed by Sung [S.H. Sung, On the strong convergence for weighted sums of ρ∗-mixing random variables,Stat. Pap., 54:773–781, 2013]. In addition, the two obtained lemmas in this paper improve the corresponding ones ofSung in the above-mentioned paper and [S.H. Sung, On the strong convergence for weighted sums of random variables,Stat. Pap., 52:447–454, 2011].

MSC: 60F15

Keywords: complete convergence, weighted sums, ρ∗-mixing random variables

1 Introduction

Let {Xk, k ∈ N} be a sequence of random variables defined on a probability space (Ω,F ,P) and define theσ-algebras

Fmn = σ(Xk, n � k � m), n � m � ∞.

As usual, for a σ-algebra F , we denote by L2(F) the class of all F-measurable random variables with finitesecond moment.

The following concept of ρ∗-mixing random variables was introduced by Moore [10].DEFINITION 1. A sequence of random variables {Xn, n � 1} is called ρ∗-mixing if for some integer k � 1,the mixing coefficient

ρ∗(k) = sup sup{Corr(X,Y ): X ∈ L2(FS), Y ∈ L2(FT )

}< 1,

where FS = σ{Xi, i ∈ S}, and the outside supremum is taken over all pairs of nonempty finite sets S, T ofintegers such that min{|s− t|, s ∈ S, t ∈ T} � k.1 Corresponding author.2 The research of author was partially supported by the Humanities and Social Sciences Foundation for the Youth Scholars of Ministryof Education of China (12YJCZH217), the Key NSF of Anhui Educational Committee (KJ2014A255), and the Natural ScienceFoundation of Anhui Province (1308085MA03).

220

0363-1672/14/5402-0220 c© 2014 Springer Science+Business Media New York

A note on the rates of convergence for weighted sums of ρ∗-mixing random variables 221

An array of random variables {Xnk, k ∈ N, n ∈ N} is said to be rowwise ρ∗-mixing if, for every n ∈ N,{Xnk, k ∈ N} is a ρ∗-mixing sequence of random variables.

Bradley [2] was the first who introduced the concept of ρ∗-mixing random variables to limit theorems.Since the article of Bradley [2], many authors studied the convergence properties for sequences or arrays ofρ∗-mixing random variables. We refer the reader to [1, 2, 3, 4, 5, 6, 9, 11, 14, 15, 17, 18, 19, 20, 21].

A sequence of random variables {Un, n ∈ N} is said to converge completely to a constant a if, for anyε > 0,

∞∑n=1

P(|Un − a| > ε

)< ∞.

This notion was given firstly by Hsu and Robbins [7].In view of the Borel–Cantelli lemma, the above result implies that Un → a almost surely. Therefore,

the complete convergence is a very important tool in establishing the almost sure convergence of sums andweighted sums of random variables.

Sung [13] obtained the following complete convergence result for weighted sums of negatively associated(NA) (see [8]) random variables.

Theorem A. Let {X, Xn, n � 1} be a sequence of identically distributed NA random variables, and let{ani, 1 � i � n, n � 1} be an array of constants satisfying

n∑i=1

|ani|α = O(n) (1.1)

for some 0 < α � 2. Let bn = n1/α(log n)1/γ for some γ > 0. Furthermore, suppose that EX = 0 when1 < α � 2. If

E|X|α < ∞ for α > γ,

E|X|α log |X| < ∞ for α = γ,

E|X|γ < ∞ for α < γ, (1.2)

then∞∑n=1

n−1P

(max

1�m�n

∣∣∣∣∣m∑i=1

aniXi

∣∣∣∣∣ > bnε

)< ∞ for all ε > 0. (1.3)

Zhou et al. [20] partially extended Theorem A to ρ∗-mixing random variables as follows.

Theorem B. Let {Xn, n � 1} be a sequence of identically distributed ρ∗-mixing random variables, and let{ani, 1 � i � n, n � 1} be an array of constants satisfying

n∑i=1

|ani|max{α,γ} = O(n) (1.4)

for some 0 < α � 2 and γ > 0 with α �= γ. Let bn = n1/α(log n)1/γ . If EX1 = 0 for 1 < α � 2 and (1.2)holds for α �= γ, then (1.3) holds.

As Sung [14] pointed out, Theorem B extends only the case α > γ of Theorem A to ρ∗-mixing randomvariables. Zhou et al. [20] left an open problem whether the case α = γ of Theorem A holds for ρ∗-mixingrandom variables. Sung [14] solved this problem and obtained the following result.

Lith. Math. J., 54(2):220–228, 2014.

222 Y. Wu, S.H. Sung, and A. Volodin

Theorem C. Let {Xn, n � 1} be a sequence of identically distributed ρ∗-mixing random variables, andlet {ani, 1 � i � n, n � 1} be an array of constants satisfying (1.1) for some 0 < α � 2. Let bn =n1/α(log n)1/α. If EX1 = 0 for 1 < α � 2 and E|X1|α log(1 + |X1|) < ∞, then (1.3) holds.

Remark 1. Sung [14, Remark 2.2] presented an open problem whether the case α < γ of Theorem A remainstrue for ρ∗-mixing random variables?

In this work, we shall study the complete convergence result for weighted sums of ρ∗-mixing randomvariables and solve the above problem.

Throughout this paper, the symbol C always stands for a generic positive constant, which may differ fromone place to another, and log x = loge x. For a finite set A, the symbol �(A) denotes the number of elementsof A.

2 Preliminaries

We first state some lemmas, which will be used in the proofs of our main results.

Lemma 1. (See [15].) Suppose that N is a positive integer, 0 � r < 1, and q � 2. Then there exists a positiveconstant C = C(N, r, q) such that the following statement holds:

If {Xk, k � 1} is a sequence of random variables such that ρ∗N � r and such that EXk = 0 andE|Xk|q < ∞ for every k � 1, then, for all n � 1,

E max1�j�n

∣∣∣∣∣j∑

k=1

Xk

∣∣∣∣∣q

� C

{n∑

k=1

E|Xk|q +(

n∑k=1

EX2k

)q/2}.

Second, we present the following lemma, which improves Lemma 2.2 and Lemma 2.3 of [13].

Lemma 2. Let {ani, 1 � i � n, n � 1} be an array of constants satisfying (1.1) for some α > 0, and X bea random variable. Let bn = n1/α(log n)1/γ for some γ > 0. Then

∞∑n=2

n−1b−αn

n∑i=1

E|aniX|αI(|aniX| > bn)�

⎧⎪⎨⎪⎩CE|X|α for α > γ,

CE|X|α log(1 + |X|) for α = γ,

CE|X|γ for α < γ.

Proof. Let

L =

∞∑n=2

n−1b−αn

n∑i=1

E|aniX|αI(|aniX| > bn).

Without loss of generality, we may assume that∑n

i=1 |ani|α � n. Then we have

L =

∞∑n=2

n−2(log n)−α/γn∑

i=1

E|aniX|αI(|ani|α|X|α > n(log n)α/γ)

�∞∑n=2


i=1

E|aniX|αI((

n∑i=1

|ani|α)|X|α > n(log n)α/γ

)

=

∞∑n=2


i=1

E|aniX|αI(|X|α > n(log n)α/γ

(n∑

i=1

|ani|α)−1)


�∞∑n=2


i=1

E|aniX|αI(|X| > (log n)1/γ)

�∞∑n=2

n−1(log n)−α/γE|X|αI(|X| > (log n)1/γ)

=

∞∑n=2

n−1(log n)−α/γ∞∑

m=n

E|X|αI(logm < |X|γ � log(m+ 1))

=

∞∑m=2

E|X|αI(logm < |X|γ � log(m+ 1)) m∑n=2

n−1(log n)−α/γ .

Observing that

m∑n=2

n−1(log n)−α/γ �

⎧⎪⎨⎪⎩C for α > γ,

C log logm for α = γ,

C(logm)1−α/γ for α < γ,

we can get

L �


CE|X|α log |X| for α = γ,


The proof of Lemma 2 is completed. ��

Remark 2. Noting that

∞∑n=2

n−1n∑

i=1

P(|aniX| > bn

)�

∞∑n=2

n−1b−αn

n∑i=1

E|aniX|αI(|aniX| > bn).

Therefore, we know that this lemma improves Lemma 2.2 and Lemma 2.3 of Sung [13]. In addition, themethod used in this paper is novel and much simpler than that in [13].

The following lemma plays an essential role in the proof of our main results, which complements Lemma 2.3of Sung [14].

Lemma 3. Let {ani, i � 1, n � 1} be an array of constants satisfying (1.1) for some α > 0, and X bea random variable. Let bn = n1/α(log n)1/γ for some γ > 0. If q > max{α, γ}, then

∞∑n=2

n−1b−qn

n∑i=1

E|aniX|qI(|aniX| � bn)�




(2.1)

Proof. Let

I =

∞∑n=2

n−1b−qn

n∑i=1

E|aniX|qI(|aniX| � bn).

Lith. Math. J., 54(2):220–228, 2014.


For j � 1 and n � 2, let

Inj ={1 � i � n: n1/α(j + 1)−1/α < |ani| � n1/αj−1/α

}.

Then {Inj , j � 1} are disjoint, and⋃

j�1 Inj = {1 � i � n: 0 < |ani| � n1/α} since∑n

i=1 |ani|α � n.Note that, for all k � 1, we have

n �n∑

i=1

|ani|α =

∞∑j=1

∑i∈Inj

|ani|α �∞∑j=1

�(Inj)n (j + 1)−1

�∞∑j=k

�(Inj)n (j + 1)−1 =

∞∑j=k

�(Inj)n (j + 1)−q/α(j + 1)q/α−1

�∞∑j=k

�(Inj)n (j + 1)−q/α(k + 1)q/α−1.

Hence, for all k � 1, we have∞∑j=k

�(Inj)j−q/α � C(k + 1)1−q/α. (2.2)

Then

I =

∞∑n=2

n−1−q/α(log n)−q/γn∑

i=1

|ani|qE|X|qI(|aniX| � n1/α(log n)1/γ)

=

∞∑n=2

n−1−q/α(log n)−q/γ∞∑j=1

∑i∈Inj

|ani|qE|X|qI(|aniX| � n1/α(log n)1/γ)

�∞∑n=2

n−1−q/α(log n)−q/γ∞∑j=1

�(Inj)nq/αj−q/αE|X|qI(|X| � (j + 1)1/α(log n)1/γ

)

�∞∑n=2

n−1(log n)−q/γ∞∑j=1

�(Inj)j−q/αE|X|qI(|X| � (log n)1/γ

)

+

∞∑n=2

n−1(log n)−q/γ∞∑j=1

�(Inj)j−q/α

×j∑

k=1

E|X|qI(k1/α(log n)1/γ < |X| � (k + 1)1/α(log n)1/γ)

=: I∗ + I∗∗.

If α > γ, by (2.2) and q > α we have

I∗ � C

∞∑n=2

n−1(log n)−q/γE|X|qI(|X|γ � log n)


� C

∞∑n=2

n−1(log n)−α/γE|X|αI(|X|γ � log n)

� CE|X|α∞∑n=2

n−1(log n)−α/γ � CE|X|α.

If α � γ, by (2.2) and q > γ we have

I∗ � C

∞∑n=2

n−1(log n)−q/γE|X|qI(|X|γ � log n)

= C

∞∑n=2

n−1(log n)−q/γn∑

m=2

E|X|qI(log(m− 1) < |X|γ � logm)

= C

∞∑m=2

E|X|qI(log(m− 1) < |X|γ � logm) ∞∑n=m

n−1(log n)−q/γ

� C

∞∑m=2

(logm)1−q/γE|X|qI(log(m− 1) < |X|γ � logm)� CE|X|γ .

Next, we consider I∗∗. By (2.2) we have

I∗∗ =∞∑n=2

n−1(log n)−q/γ∞∑k=1

E|X|qI(k1/α(log n)1/γ < |X| � (k + 1)1/α(log n)1/γ) ∞∑j=k

�(Inj) j−q/α

� C

∞∑n=2

n−1(log n)−q/γ∞∑k=1

(k + 1)1−q/αE|X|qI(k1/α(log n)1/γ < |X| � (k + 1)1/α(log n)1/γ)

� C

∞∑n=2

n−1(log n)−α/γ∞∑k=1

E|X|αI(k1/α(log n)1/γ < |X| � (k + 1)1/α(log n)1/γ)

= C

∞∑n=2

n−1(log n)−α/γE|X|αI(|X| > (log n)1/γ)

(similarly to the proof of Lemma 2)

�




Therefore,

I � I∗ + I∗∗ �




The proof is completed. ��Remark 3. Clearly, Lemma 2.3 of Sung [14] is a special case of Lemma 3 in this paper (α = γ). It is worthpointing out that, in [14], it is required that ani = 0 or |ani| > 1. Here, we do not require these extra conditions.

Lith. Math. J., 54(2):220–228, 2014.


3 Main result

Now we state the following result, which solves the open problem described in Remark 1.Theorem 1. Let {Xn, n � 1} be a sequence of identically distributed ρ∗-mixing random variables, andlet {ani, 1 � i � n, n � 1} be an array of constants satisfying (1.1) for some 0 < α � 2. Let bn =n1/α(log n)1/γ . If EX1 = 0 for 1 < α � 2 and (1.2) holds for α < γ, then (1.3) holds.

Proof. Without loss of generality, we may assume that∑n

i=1 |ani|α � n. For fixed n � 1, let

Yni = aniXiI(|aniXi| � bn

), Zni = aniXi − Yni.

Then

∞∑n=1

n−1P

(max1�j�n

∣∣∣∣∣j∑

i=1

aniXi

∣∣∣∣∣ > bnε

)

� 1 +

∞∑n=2

n−1n∑

i=1

P(|aniXi| > bn

)+

∞∑n=2

n−1P

(max1�j�n

∣∣∣∣∣j∑

i=1

Yni

∣∣∣∣∣ > bnε

)

=: 1 + J1 + J2.

To prove (1.3), we need only to show that J1 < ∞ and J2 < ∞. By Lemma 2 we get

J1 � C

∞∑n=2

n−1b−αn

n∑i=1

|ani|αE|X1|αI(|aniX1| > bn

)� CE|X1|γ < ∞.

Next, we prove J2 < ∞. We first show that

b−1n max

1�j�n

∣∣∣∣∣j∑

i=1

EYni

∣∣∣∣∣ → 0 as n → ∞. (3.1)

Clearly, E|X|γ < ∞ and α < γ imply E|X|α < ∞. If 0 < α � 1, by∑n

i=1 |ani|α � n we have

b−1n max

1�j�n

∣∣∣∣∣j∑

i=1

EYni

∣∣∣∣∣ � b−1n

n∑i=1

|ani|E|Xi|I(|aniXi| � bn

)� b−α

n

n∑i=1

|ani|αE|Xi|αI(|aniXi| � bn

)� (log n)−α/γE|X1|α → 0 as n → ∞.

If 1 < α � 2, by EX1 = 0 and∑n

i=1 |ani|α � n we also have

b−1n max

1�j�n

∣∣∣∣∣j∑

i=1

EYni

∣∣∣∣∣ � b−1n

n∑i=1

|ani|E|Xi|I(|aniXi| > bn

)� b−α

n

n∑i=1

|ani|αE|Xi|αI(|aniXi| > bn

)� (log n)−α/γE|X1|α → 0 as n → ∞.

To sum up, we prove that (3.1) holds for 0 < α � 2. Hence, when n is sufficiently large, we have

max1�j�n

∣∣∣∣∣j∑

i=1

EYni

∣∣∣∣∣ < bnε

2.


Let q > max{2, 2γ/α}. By the Markov inequality and Lemma 1 we have

J2 �∞∑n=2

n−1P

(max1�j�n

∣∣∣∣∣j∑

i=1

(Yni −EYni)

∣∣∣∣∣ > bnε

2

)� C

∞∑n=2

n−1b−qn E max

1�j�n

∣∣∣∣∣j∑

i=1

(Yni −EYni)

∣∣∣∣∣q

� C

∞∑n=2

n−1b−qn

n∑i=1

E|Yni|q + C

∞∑n=2

n−1b−qn

(n∑

i=1

EY 2ni

)q/2

=: J3 + J4.

By Lemma 3 and α < γ we get

J3 � C

∞∑n=2

n−1b−qn

n∑i=1

|ani|qE|Xi|qI(|aniXi| � bn

)� CE|X1|γ � ∞.

Finally, we prove J4 < ∞. By α � 2,∑n

i=1 |ani|α � n, and q > 2γ/α we have

J4 � C

∞∑n=2

n−1

(b−αn

n∑i=1

|ani|αE|Xi|αI(|aniXi| � bn

))q/2

� C

∞∑n=2

n−1(log n)−αq/(2γ)(E|X1|α

)q/2< ∞.

The proof is completed. ��

Remark 4. As Sung [14] pointed out, the crucial tool of the proof of Theorem 1 is the Rosenthal-type inequalityfor maximum partial sums of ρ∗-mixing random variables. For m-dependent random variables and ϕ-mixingrandom variables satisfying

∑∞n=1 ϕ

1/2(n) < ∞, the Rosenthal-type inequality for maximum partial sumsalso holds (see [12] and [16], respectively). Therefore, Theorem 1 also holds for m-dependent random vari-ables and ϕ-mixing random variables satisfying

∑∞n=1 ϕ

1/2(n) < ∞.

References

1. J. An and D.M. Yuan, Complete convergence of weighted sums for ρ̃-mixing sequence of random variables, Stat.Probab. Lett., 78:1466–1472, 2008.

2. R.C. Bradley, Equivalent mixing conditions for random fields, Technical Report No. 336, Center for StochasticProcesses, Department of Statistics, University of North Carolina, Chapel Hill, 1990.

3. G.H. Cai, Marcinkiewicz strong laws for linear statistics of ρ̃-mixing sequences of random variables, An. Acad. Bras.Ciênc., 78:615–621, 2006.

4. G.H. Cai, Moment inequality and complete convergence of ρ̃-mixing sequences, J. Syst. Sci. Math. Sci., 28:251–256,2008 (in Chinese).

5. S.X. Gan, Almost sure convergence for ρ̃-mixing random variable sequences, Stat. Probab. Lett., 67:289–298, 2004.

6. M.L. Guo and D.J. Zhu, Equivalent conditions of complete moment convergence of weighted sums for ρ∗-mixingsequence of random variables, Stat. Probab. Lett., 83:13–20, 2013.

7. P.L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA, 33:25–31,1947.

Lith. Math. J., 54(2):220–228, 2014.


8. K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Stat., 11:286–295,1983.

9. A. Kuczmaszewska, On complete convergence for arrays of rowwise dependent random variables, Stat. Probab.Lett., 77:1050–1060, 2007.

10. C.C. Moore, The degree of randomness in a stationary time series, Ann. Math. Stat., 34:1253–258, 1963.

11. M. Peligrad and A. Gut, Almost-sure results for a class of dependent random variables, J. Theor. Probab., 12:87–104,1999.

12. Q.M. Shao, A moment inequality and its application, Acta Math. Sin., Chin. Ser., 31:736–747, 1988 (in Chinese).

13. S.H. Sung, On the strong convergence for weighted sums of random variables, Stat. Pap., 52:447–454, 2011.

14. S.H. Sung, On the strong convergence for weighted sums of ρ∗-mixing random variables, Stat. Pap., 54:773–781,2013.

15. S. Utev and M. Peligrad, Maximal inequalities and an invariance principle for a class of weakly dependent randomvariables, J. Theor. Probab., 16:101–115, 2003.

16. X.J. Wang, S.H. Hu, W.Z. Yang, and Y. Shen, On complete convergence for weighted sums of ϕ-mixing randomvariables, J. Inequal. Appl., 2010, doi:10.1155/2010/372390.

17. X.J. Wang, X.Q. Li, W.Z. Yang, and S.H. Hu, On complete convergence for arrays of rowwise weakly dependentrandom variables, Appl. Math. Lett., 25:1916–1920, 2012.

18. Q.Y. Wu and Y.Y. Jiang, Some strong limit theorems for ρ̃-mixing sequences of random variables, Stat. Probab.Lett., 78:1017–1023, 2008.

19. Y.F. Wu, C.H. Wang, and A. Volodin, Limiting behavior for arrays of rowwise ρ∗-mixing random variables, Lith.Math. J., 52:214–221, 2012.

20. X.C. Zhou, C.C. Tan, and J.G. Lin, On the strong laws for weighted sums of ρ∗-mixing random variables, J. Inequal.Appl., 2011, doi:10.1155/2011/157816.

21. M.H. Zhu, Strong laws of large numbers for arrays of rowwise ρ∗-mixing random variables, Discrete Dyn. Nat. Soc.,2007, Article ID 74296, 6pp., 2007.

A note on the rates of convergence for weighted sums of ρ * -mixing random variables

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Transcript of A note on the rates of convergence for weighted sums of ρ * -mixing random variables