Complete Convergence of Weighted Sums for Asymptotically Almost Negatively Associated Sequences ()
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).