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On complete convergence for weighted sums of martingale-difference random fields

Journal of Inequalities and Applications20132013:473

https://doi.org/10.1186/1029-242X-2013-473

  • Received: 21 June 2013
  • Accepted: 23 September 2013
  • Published:

Abstract

Let { a n , i , n Z + d , i n } be an array of real numbers, and let { X i , i Z + d } be the martingale differences with respect to { G n , n Z + d } satisfying E ( E ( X | G k ) | G m ) = E ( X | G k m ) a.s., where k m denotes componentwise minimum, { G k , k Z + d } is a family of σ-algebras such that k n , G k G n G , and X is any integrable random variable defined on the initial probability space. The aim of this paper is to obtain some results concerning complete convergence of weighted sums i n a n , i X i .

MSC:60F05, 60F15.

Keywords

  • complete convergence
  • weighted sums
  • martingale difference
  • maximal moment inequality
  • σ-algebra

1 Introduction

The concept of complete convergence for sums of independent and identically distributed random variables was introduced by Hsu and Robbins [1] as follows: A sequence of random variables { X n } is said to be completely to a constant c if
n = 1 P ( | X n c | > ϵ ) < for all  ϵ > 0 .

This result has been generalized and extended to the random fields { X n , n Z + d } of random variables. For example, Fazekas and Tómács [2] and Czerebak-Mrozowicz et al. [3] for fields of pairwise independent random variables, and Gut and Stadtmüller [4] for random fields of i.i.d. random variables.

Let Z + be the set of positive integers. For fixed d Z + , set Z + d = { n = ( n 1 , n 2 , , n d ) : n i Z + , i = 1 , 2 , , d } with coordinatewise partial order, ≤, i.e., for m = ( m 1 , m 2 , , m d ) , n = ( n 1 , n 2 , , n d ) Z + d , m n if and only if m i n i , i = 1 , 2 , , d . For n = ( n 1 , n 2 , , n d ) Z + d , let | n | = i = 1 d n i . For a field { a n , n Z + d } of real numbers, the limit superior is defined by inf r 1 sup | n | r a n and is denoted by lim sup | n | a n .

Note that | n | is equivalent to max { n 1 , n 2 , , n d } , which is weaker than the condition min { n 1 , n 2 , , n d } when d 2 .

Let { X n , n Z + d } be a field of random variables, and let { a n , k , n Z + d , k n } be an array of real numbers. The weighted sums k n a n , k X k can play an important role in various applied and theoretical problems, such as those of the least squares estimators (see Kafles and Bhaskara Rao [5]) and M-estimates (see Rao and Zhao [6]) in linear models, the nonparametric regression estimators (see Priestley and Chao [7]), etc. So, the study of the limiting behavior of the weighted sums is very important and significant (see Chen and Hao [8]).

Now, we consider the notion of martingale differences. Let { G k , k Z + d } be a family of σ-algebras such that
G k G n G , k n ,
and for any integrable random variable X defined on the initial probability space,
E ( E ( X | G k ) | G m ) = E ( X | G k m ) a.s. ,
(1.1)

where k m denotes the componentwise minimum.

An { G k , k Z + d } -adapted, integrable process { Y k , k Z + d } is called a martingale if and only if
E ( Y n | G m ) = Y m n a.s.
Let us observe that for martingale { ( Y n , G n ) , n Z + d } , the random variables
X n = a { 0 , 1 } d ( 1 ) i = 1 d a i Y n a ,

where a = ( a 1 , a 2 , , a d ) and n Z + d , are martingale differences with respect to { G n , n Z + d } (see Kuczmaszewska and Lagodowski [9]).

For the results concerning complete convergence for martingale arrays obtained in the one-dimensional case, we refer to Lagodowski and Rychlik [10], Elton [11], Lesigne and Volny [12], Stoica [13] and Ghosal and Chandra [14]. Recently, complete convergence for martingale difference random fields was proved by Kuczmaszewska and Lagodowski [9].

The aim of this paper is to obtain some results concerning complete convergence of weighted sums i n a n , i X i , where { a n , i , n Z + d , i n } is an array of real numbers, and { X i , i Z + d } is the martingale differences with respect to { G n , n Z + d } satisfying (1.1).

2 Results

The following moment maximal inequality provides us a useful tool to prove the main results of this section (see Kuczmaszewska and Lagodowski [9]).

Lemma 2.1 Let { ( Y n , G n ) , n Z + d } be a martingale, and let { ( X n , G n ) , n Z + d } be the martingale differences corresponding to it. Let q > 1 . There exists a finite and positive constant C depending only on q and d such that
E ( max k n | Y k | q ) C E ( k n X k 2 ) q / 2 .
(2.1)

Let us denote G i = σ { G j : j < i } . Now, we are ready to formulate the next result.

Theorem 2.2 Let { a n , i , n Z + d , i n } be an array of real numbers, and let { X n , n Z + d } be the martingale differences with respect to { G n , n Z + d } satisfying (1.1). For α p > 1 , p > 1 and α > 1 2 , we assume that
  1. (i)

    n | n | α p 2 i n P { | a n , i X i | > | n | α } < ,

     
  2. (ii)

    n | n | α ( p q ) 3 + q / 2 i n | a n , i | q E ( | X i | q I [ | a n , i X i | | n | α ] ) < for q 2 ,

     
(ii)′ n | n | α ( p q ) 2 i n | a n , i | q E ( | X i | q I [ | a n , i X i | | n | α ] ) < for 1 < q < 2 and
  1. (iii)

    n | n | α p 2 P { max j n | i j E ( a n , i X i I [ | a n , i X i | | n | α ] | G i ) | > ϵ | n | α } < for all ϵ > 0 .

     
Then we have
n | n | α p 2 P { max j n | S j | > ϵ | n | α } < for all  ϵ > 0 ,
(2.2)

where S n = 1 i n a n , i X i .

Proof Let us notice that the series n | n | α p 2 is finite, then (2.2) always holds. Therefore, we consider only the case such that n | n | α p 2 is divergent. Let X n , i = X i I [ | a n , i X i | | n | α ] , X n , i = X n , i E ( X n , i | G i ) and S n , j = i j a n , i X n , i .

Then
n | n | α p 2 P { max j n | S j | > ϵ | n | α } n | n | α p 2 P { | a n , i X i | > | n | α } + n | n | α p 2 P { max j n | i j a n , i X i I [ | a n , i X i | | n | α ] | > ϵ | n | α } n | n | α p 2 i n P { | a n , i X i | > | n | α } + n | n | α p 2 P { max j n | i j ( a n , i X i I [ | a n , i X i | | n | α ] E ( a n , i X i I [ | a n , i X i | | n | α ] | G i ) ) | > ϵ 2 | n | α } + n | n | α p 2 P { max j n | i j E ( a n , i X i I [ | a n , i X i | | n | α ] | G i ) | > ϵ 2 | n | α } = I 1 + I 2 + I 3 .
Clearly, I 1 < by (i), and I 3 < by (iii). It remains to prove that I 2 < . Thus, the proof will be completed by proving that
n | n | α p 2 P { max j n | S n , j | > ϵ | n | α } < .
To prove it, we first observe that { ( S n , j , G j ) , j n } is a martingale. In fact, if i > j , then G i j G i and by (1.1), we have
E ( a n , i X n , i | G j ) = E ( a n , i X n , i E ( a n , i X n , i | G i ) | G i ) = E ( E ( a n , i X n , i E ( a n , i X n , i | G i ) | G i ) | G j ) = E ( a n , i X n , i E ( a n , i X n , i | G i ) | G i j ) = 0 .
Then, by the Markov inequality and Lemma 2.1, there exists some constant C such that
P { max j n | S n , j | > ϵ | n | α } C E ( max j n | S n , j | q ) | n | α q C | n | α q E ( i n a n , i 2 X n , i 2 ) q / 2 = I 4 .
Case q 2 ; we get
I 4 C | n | α q | n | q / 2 1 i n E | a n , i X n , i | q C | n | q / 2 1 α q i n E ( | a n , i X i | q I [ | a n , i X i | | n | α ] ) .
Note that the last estimation follows from the Jensen inequality. Thus, we have
n | n | α p 2 P { max j n | S n , j | > ϵ | n | α } C n | n | α p 3 q ( α 1 / 2 ) i n E ( | a n , i X i | q I [ | a n , i X i | | n | α ] ) <

by assumption (ii).

Case 1 < q < 2 ; we get
I 4 C | n | α q i n E | a n , i X n , i | q C | n | α q i n E ( | a n , i X i | q I [ | a n , i X i | | n | α ] ) .
Therefore, for 1 < q < 2 , we obtain
n | n | α p 2 P { max j n | S n , j | > ϵ | n | α } C n | n | α ( p q ) 2 i n E ( | a n , i X i | q I [ | a n , i X i | | n | α ] ) <

by assumption (ii)′. Thus, I 2 < for all q > 1 , and the proof of Theorem 2.2 is complete. □

Corollary 2.3 Let { a n , i , n Z + d , i n } be an array of real numbers. Let { X n , n Z + d } be martingale differences with respect to { G n , n Z + d } satisfying (1.1), and E X n = 0 for n Z + d . Let p 1 , α > 1 2 and α p > 1 . Assume that (i) and for some q > 1 , (ii) or (ii)′ hold respectively. If
max j n i j E ( a n , i X i I [ | a n , i X i | | n | α ] | G i ) = o ( | n | α ) ,
(2.3)

then (2.2) holds.

Proof It is easy to see that (2.3) implies (iii). We omit details that prove it. □

The following corollary shows that assumption (iii) in Theorem 2.2 is natural, and in the case of independent random fields, it reduces to the known one.

Corollary 2.4 Let { a n , i , n Z + d , i n } be an array of real numbers. Let { X n , n Z + d } be a field of independent random variables such that E X n = 0 for n Z + d . Let p 1 , α > 1 2 and α p > 1 . Assume that (i) and for some q > 1 , (ii) or (ii)′ hold respectively. If
1 | n | α max j n i j E ( a n , i X i I [ | a n , i X i | | n | α ] ) 0 as  | n | ,
(2.4)

then (2.2) holds.

Proof Since { X n , n Z + d } is a field of independent random variables, we have
1 | n | α max j n i j E ( a n , i X i I [ | a n , i X i | | n | α ] | G i ) = 1 | n | α max j n i j E ( a n , i X i I [ | a n , i X i | | n | α ] ) .

Now, it is easy to see that (2.4) implies (iii) of Theorem 2.2. Thus, by Theorem 2.2, result (2.2) follows. □

Remark Theorem 2.2 and Corollary 2.4 are extensions of Theorem 4.1 and Corollary 4.1 in Kuczmaszewska and Lagodowski [9] to the weighted sums case, respectively.

Corollary 2.5 Let { a n , i , n Z + d , 1 i n } be an array of real numbers. Let { X n , n Z + d } be the martingale differences with respect to { G n , n Z + d } satisfying (1.1) and E X n = 0 . Let p 1 , α > 1 2 and α p > 1 and E | X n | 1 + λ n < for λ n with 0 < λ n < 1 for n Z + d . If
n | n | α p 2 | n | α ( 1 + λ n ) i n | a n , i | 1 + λ n E | X i | 1 + λ n < ,
(2.5)
max 1 j n i j E ( | a n , i X i | I [ | a n , i X i | | n | α ] | G i ) = o ( | n | α ) ,
(2.6)

then (2.2) holds.

Proof If the series n | n | α p 2 < , then (2.2) always holds. Hence, we only consider the case n | n | α p 2 = . It follows from (2.5) that
| n | α ( 1 + λ n ) i n | a n , i | 1 + λ n E | X i | 1 + λ n < 1 .
By (2.5) and the Markov inequality,
n | n | α p 2 P ( | a n , i X i | > | n | α ) n | n | α p 2 | n | α ( 1 + λ n ) i n | a n , i | 1 + λ n E | X i | 1 + λ n < ,
(2.7)

which satisfies (i) of Theorem 2.2.

As the proof of Corollary 2.3, (2.6) implies (iii) of Theorem 2.2.

It remains to show that Theorem 2.2(ii) or (ii)′ is satisfied.

For 1 < q < 2 , take 1 + λ n < q . Then we have
n | n | α ( p q ) 2 i n | a n , i | q E ( | X i | q I [ | a n , i X i | | n | α ] ) n | n | α p 2 | n | α ( 1 + λ n ) | n | α q + α ( 1 + λ n ) | n | α q α ( 1 + λ n ) i n | a n , i | 1 + λ n E | X i | 1 + λ n = n | n | α p 2 | n | α ( 1 + λ n ) i n | a n , i | 1 + λ n E | X i | 1 + λ n < by (2.5) ,

which satisfies Theorem 2.2(ii)′. Hence, the proof is complete. □

Corollary 2.6 Let { a n , i , n Z + d , 1 i n } be an array of real numbers, and let { X n , n Z + d } be the martingale differences with respect to { G n , n Z + d } satisfying (1.1), E X n = 0 and E | X n | p < for 1 < p < 2 . Let α > 1 2 , α p > 1 and 1 < p < 2 . If
1 i n | a n , i | p E | X i | p = O ( | n | δ ) for  0 < δ < 1 ,
(2.8)

and Theorem  2.2(iii) hold, then (2.2) holds.

Proof By (2.8) and the Markov inequality,
n | n | α p 2 i n P ( | a n , i X i | > | n | α ) n | n | α p 2 i n | a n , i | p E | X i | p | n | α p C n | n | 2 + δ < .
(2.9)
By taking q < p , we have
n | n | α ( p q ) 2 i n | a n , i | q E ( | X i | q I [ | a n , i X i | ϵ | n | α ] ) n | n | 2 i n | a n , i | p E | X i | p C n | n | 2 + δ < .
(2.10)

Hence, by (2.9) and (2.10), conditions (i) and (ii)′ in Theorem 2.2 are satisfied, respectively.

To complete the proof, it is enough to note that by E X n = 0 for n Z + d and by (2.8), we get for j n
| n | α i j | a n , i | E | X i | I [ | a n , i X i | ϵ | n | α ] 0 as  | n | .
(2.11)

Hence, the proof is complete. □

Corollary 2.7 Let { X n , n Z + d } be the martingale differences with respect to { G n , n Z + d } satisfying (1.1), let E X n = 0 and E | X n | p < for 1 < p < 2 and be stochastically dominated by a random variable X, i.e., there is a constant D such that P ( | X n | > x ) D P ( | X | > x ) for all x 0 and n Z + d . Let { a n , i , n Z + d , i n } be an array of real numbers satisfying
i n | a n , i | p = O ( | n | δ ) for  0 < δ < 1 .
(2.12)

If Theorem  2.2(iii) holds, then (2.2) holds.

Proof From (2.12), (2.8) follows. Hence, by Corollary 2.6, we obtain (2.2). □

Remark Linear random fields are of great importance in time series analysis. They arise in a wide variety of context. Applications to economics, engineering, and physical science are extremely broad (see Kim et al. [15]).

Let Y k = i 1 a k + i X i , where { a i , i Z + d } is a field of real numbers with i 1 | a i | < , and { X i , i Z + d } is a field of the martingale difference random variables.

Define a n , i = 1 k n a i + k . Then we have
1 k n Y k = 1 k n i 1 a i + k X i = i 1 1 k n a i + k X i = i 1 a n , i X i .

Hence, we can use the above results to investigate the complete convergence for linear random fields.

Declarations

Authors’ Affiliations

(1)
Division of Mathematics and Informational Statistics, Wonkwang University, Jeonbuk, 570-749, Korea

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© Ko; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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