Applied Mathematics
Vol.08 No.11(2017), Article ID:80691,9 pages
10.4236/am.2017.811120
Complete Convergence of Weighted Sums for Asymptotically Almost Negatively Associated Sequences
Jun An
College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, China
Copyright © 2017 by author and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: August 25, 2017; Accepted: November 26, 2017; Published: November 29, 2017
ABSTRACT
For weighted sums of asymptotically almost negatively associated (AANA) random variables sequences, we use the Rosenthal type moment inequalities and prove the Marcinkiewicz-Zygmund type complete convergence and obtain the complete convergence rates. Our results extend some known ones.
Keywords:
Asymptotically Almost Negatively Associated, Weighted Sums, Complete Convergence
1. Introduction
A sequence of random variables is said to be asymptotically almost negatively associated (AANA, in short) if there exists a nonegative sequence as such that
(1.1)
for all and for all coordinate wise nondecreasing continuous functions and whenever the variances exist. is said to be the mixing coefficients of .
Chandra and Ghosal [1] firstly introduced this concept, and gave a following example. Let , where are independent random variables with common distribution , then is an AANA sequence. At the same time, the Kolmogorov type inequality and strong law of large numbers (SLLN) were proved.
From then on, many authors have studied the various limit properties for AANA sequences. For example, Chandra and Ghosal [2] [3] obtained the almost sure convergence of weighted average, Kim, Ko and Lee [4] established the Hajek-Renyi type inequalities and Marcinkiewicz-Zygmund type SLLN, Cai [5] investigated the complete convergence of weighted sums, Yuan and An [6] got the Rosenthal type inequalities, convergence, complete convergence and Marcinkiewicz-Zygmund type SLLN, Wang, Hu and Yang [7] obtained the complete convergence and SLLN, etc. and so on. We see the following theorems.
Theorem A. (Kim, Ko and Lee [4] ) Let be a sequence of real numbers with and let be a sequence of identically distributed, mean zero AANA random variables with . If , then
(1.2)
Theorem A generalizes the Marcinkiewicz-Zygmund SLLN (Chow and Teicher [1] , or Gut [8] ) for the independent identically distributed (i.i.d.) sequences to the weighted sums of AANA sequence.
Theorem B. (Cai [5] ) Let be a sequence of real numbers with , and let be a sequence of mean zero AANA random variables. Let . If , for and . Then for all ,
(1.3)
Theorem C. (Yuan and An [6] ) Let be an AANA sequence of identically distributed random variables with mixing coefficients , and suppose that for . If where , then is equivalent to
(1.4)
The main purpose of this paper is to further investigate the complete convergence, almost sure convergence and complete convergence rate of weighted sums for AANA random variable sequences. In the following sections, theorem 2.1 (Section 2) extends theorem A to some more relaxed conditions and gets a more general result. Theorem 2.2 is about complete convergence rates which extends theorem B and theorem C to the cases of weighted sums.
2. Main Results
Throughout this paper we use the following notations: denotes the indicator function, stands for a positive constant its value may be different on different places, represents the Vinogradov symbol , means defined as and denotes the norm.
Theorem 2.1. Let be a sequence of mean zero, identically
distributed AANA random variables with . Let be a sequence of real numbers satisfying . If , then
(2.1)
Remark 2.1. As we known, complete convergence leads to the almost sure convergence but its converse does not hold. So the result of theorem 2.1 is
stronger than theorem A. On the other hand, under the condition of . Thus theorem A is a corollary of
theorem 2.1.
Theorem 2.2. Let be a sequence of centered identically distributed AANA random variables with mixing coefficients , . Suppose that for . Let
be a sequence of real numbers with , if ; or if : Take , where is a positive integer number satisfying . If , then for any
(2.2)
and
. (2.3)
3. Proofs
To prove our results we need the following two lemmas.
Lemma 3.1. (Yuan and An [6] ) Let be a sequence of AANA random variables with mixing coefficients . Let be all nondecreasing (or all nonincreasing) functions, then is still a sequence of AANA random variables with mixing coefficients .
Lemma 3.2. (Yuan and An [6] ) Let be a sequence of AANA random variables with mean zero and mixing coefficients , then there exists a positive constant depending only on such that
(3.1)
for all and , and such that
(3.2)
for all and where integer number .
In particular, if , then
(3.3)
for all and .
Remark 3.1. It’s obvious that if , taking on the right hand of (3.2), the two inequalities (3.1) and (3.2) are the same.
Corollary 3.1. Under the conditions of Lemma 3.2, we have the following moment inequality
(3.4)
for all and , where integer number , .
Proof of Corollary 3.1. For , we know . Since , for large enough, there exists a positive constant such that
(3.5)
Applying the Holder inequality on the right hand of (3.5) we get
(3.6)
Thus (3.4) follows from (3.2), (3.5) and (3.6).
Proof of Theorem 2.1. Without loss of generality we may assume for all . Let
Since Lemma 3.1, and are AANA for all . It’s easy to see that
To prove (2.1) it suffices to prove completely, and completely.
By assumption and the inequality we have . For , considering two cases and
we can easily get
(3.7)
Thus, to prove completely it suffices to prove
(3.8)
By the Chebyseve inequality, and (3.3) of Lemma 3.2 we get
(3.9)
It’s easy to see that
(3.10)
For we have
(3.11)
Thus (3.8) follows from (3.9), (3.10) and (3.11). Consequently .
Since (3.7), to prove completely it suffices to prove
(3.12)
In fact, according to the Chebyshev inequality, (3.3) and assumption
,
(3.13)
From (3.12) and (3.13) we know completely. The proof of Theorem 2.1 is complete.
Proof of Theorem 2.2. Without loss of generality we assume for all . Let
.
By Lemma 2.1 we see that and are AANA for all . So
(3.14)
To prove (2.2) it suffices to prove and .
Since we have
(3.15)
By the inequality and assumption , we have . Under the condition , we consider two cases
and respectively, and we can easily get
. (3.16)
From (3.16), the Chebyshev inequality and Corollary 2.1 we know
(3.17)
By condition and the inequality
(3.18)
No matter or we have . Using assumption and the method of (3.11) we get
(3.19)
and
(3.20)
From (3.18), (3.19) and (3.20) we know .
(3.21)
Since and the inequality, we know . Therefore
(3.22)
We consider the following two cases:
1) If , then , taking the method of (3.11) we have
(3.23)
and
(3.24)
2) If , then . From and (3.22) we get
, (3.25)
and
(3.26)
Thus follows from (3.23) and (3.26), from (3.24) and (3.26). So by (3.21). Hence by (3.17). (2.2) is proved.
As for (2.3), inspired by Gut [7] (page 318_319), we have
(3.27)
Thus (2.3) follows from (2.2) and (3.27). The proof of Theorem 2.2 is completed.
Fund
This work is supported by the Projects of Science and Technology Research of Chongqing City Education Committee (KJ1307XX), and Major Social Science Commissioned Research Project of Chongqing “Research on Frontier Theory of Census Quality Assessment” (2016WT03).
Cite this paper
An, J. (2017) Complete Convergence of Weighted Sums for Asymptotically Almost Negatively Associated Sequences. Applied Mathematics, 8, 1662-1670. https://doi.org/10.4236/am.2017.811120
References
- 1. Chow, Y.S. and Teicher, H. (1997) Probability Theory: Independent, Interchangeability, Martingales. 3rd Edition, Springer, Berlin.
- 2. Chandra, T.K. and Ghosal, S. (1996) Extensions of the Strong Law of Large Numbers of Marcinkiewicz and Zygmund for Dependent Variables. Acta Mathematica Hungarica, 71, 327-336.
- 3. Chandra, T.K. and Ghosal, S. (1996) The Strong Law of Large Numbers for Weighted Averages under Dependence Assumptions. Journal of Theoretical Probability, 9, 797-809.
- 4. Kim, T.S., Ko, M.H. and Lee, I.H. (2004) On the Strong Laws for Asymptotically almost Negatively Associated Random Variables. Rocky Mountain Journal of Mathematics, 34, 979-989.
- 5. Cai, G.H. (2004) Complete Convergence for Weighted Sums of Sequences of AANA Random Variables. Glasgow Mathematical Journal, 17, 165-181.
- 6. Yuan, D.M. and An, J. (2009) Rosenthal Type Inequalities for Asymptotically Almost Negatively Associated Random Variables and Applications. Science in China Series, 52, 1887-1904.
- 7. Wang, X.J., Hu, S.H. and Yang, W.Z. (2011) Complete Convergence for Arrays of Rowwise Asymptotically Almost Negatively Associated Random Variables. Discrete Dynamics in Nature and Society, Article ID: 717126. https://doi.org/10.1155/2011/717126
- 8. Gut, A. (2005) Probability Theory: A Graducate Course. Springer, Berlin.