Inequalities for sums of adapted random fields in Banach spaces and their application to strong law of large numbers
© Dang et al.; licensee Springer. 2014
Received: 2 June 2014
Accepted: 23 October 2014
Published: 4 November 2014
We obtain inequalities for sums of adapted random fields in p-uniformly smooth Banach spaces, and we extend Kolmogorov and Marcinkiewicz-Zygmund strong laws for blockwise α-martingale difference fields.
MSC:60B11, 60B12, 60F15, 60G42.
The concept of blockwise-dependence was introduced by Móricz . Móricz’s  and Gaposhkin  showed that some properties of sequences of independent random variables can be applied to sequences consisting of independent blocks. Huan et al.  extended the strong laws of large numbers to blockwise-martingale difference arrays in Banach spaces. Recently, Móricz et al.  introduced the concept of blockwise M-dependence for a double array of random variables and established a version of the Kolmogorov SLLN for double arrays of random variables which are blockwise M-dependent. The results of Móricz and Stadtmüller and Thalmaier  were generalized by Stadtmüller and Thanh .
The aim of this paper is to investigate inequalities for sums of random fields and the strong law of large numbers of arbitrary random fields taking values in a Banach space. In Section 2, we introduce α-strong adapted random fields, α-strong∗ adapted random fields, blockwise α-martingale difference fields and prove some useful lemmas. In Section 3, inequalities for sums of α-strong adapted random fields and α-strong∗ adapted random fields in p-uniformly smooth Banach spaces are given. Section 4 contains the main results including the SLLN for a such blockwise α-martingale difference field taking values in a p-uniformly smooth Banach space, in which the results of [3, 6, 7] will be generalized.
Throughout this paper, the symbol C will denote a generic constant () which is not necessarily the same one in each appearance.
2 Preliminaries and some useful lemmas
Clearly, every real separable Banach space is 1-uniformly smooth and the real line (the same as any Hilbert space) is 2-uniformly smooth. If a real separable Banach space is p-uniformly smooth for some then it is r-uniformly smooth for all .
Let d be a positive integer, the set of all integer d-dimensional lattice points will be denoted by and the set of all positive integer d-dimensional lattice points will be denoted by . For , , denote is a d-dimensional rectangle, , , , , . We write (or ) if , ; if , and . For , let denote the greatest integer less than or equal to x, we use log+x to denote the (the logarithms are to base 2).
For nondecreasing sequences of positive integers (), for , let .
Let be a probability space, be a real separable Banach space, and be the σ-algebra of all Borel sets in . Let be a field of -valued random variables and be a field of nondecreasing sub-σ-algebras of ℱ with respect to the partial order ⪯ on such that is -measurable for all , then is said to be an adapted field.
The adapted field is said to be α-strong adapted (or strong adapted) if (or ) is -measurable for all , and .
The adapted field is said to be α-strong∗ adapted (or strong∗ adapted) if is α-strong adapted (or strong adapted) for all .
Let for all , then if , if otherwise, for all and , so is a strong∗ adapted field.
Example 2.2 Let be a field of independent random variables. Put and , so is not a field of independent random variables. If for all , then is a strong∗ adapted field.
The adapted field is said to be an α-martingale difference field if for all and .
When for all then the adapted field is said to be an M-martingale difference field.
When for all then the field is a martingale difference field which was introduced by Huan et al.  in case .
Let be a field of martingale differences, then it is strong adapted, but it is not necessarily a strong∗ adapted field.
Let be a field of m-dependence random variables with mean 0. Put and , then for all , . Therefore, is a field of M-martingale differences.
Example 2.4 Let be a field of independent random elements with mean 0. Put , then for all , . Therefore, is a field of martingale differences and a strong∗ adapted field.
Set , then is a field of α-martingale differences.
Example 2.5 Let be a field of independent random variables with mean 0. Put and , so is not a field of independent random variables. If for all , then , for all , , . Therefore, is a field of martingale differences and a strong∗ adapted field.
The adapted field is said to be a blockwise-adapted field (respectively, blockwise-α-strong adapted, blockwise-α-strong∗ adapted, blockwise-α-martingale difference field, blockwise-M-martingale difference field, blockwise-martingale difference field) with respect to the blocks if for each , is an adapted field (respectively, α-strong adapted, α-strong∗ adapted, α-martingale difference field, M-martingale difference field, martingale difference field).
Example 2.6 Let , , as in Example 2.4. Set and , , then is a blockwise-α-martingale differences and a blockwise-α-strong∗ adapted field with respect to the blocks .
To prove the main result we need the following lemmas.
Firstly, for , note that is a nonnegative sub-martingale. Applying Doob’s inequality and by (2.1), we have (2.2). We assume that (2.2) holds for , we wish to show that it holds for d.
We note that if is strong adapted, when then is a sequences of martingale differences, but when then is not necessarily a field of martingale differences, because may not be -measurable (see , Example 3.1).
and we have (2.4).
The conclusion of the lemma follows upon letting and then . □
3 Inequalities for sums of adapted random fields
The first theorem characterizes the p-uniformly smooth Banach spaces.
is p-uniformly smooth.
- (ii)There exists a positive constant C such that for all α-strong adapted random fields in we have(3.1)
Proof We first prove the implication ((i) ⇒ (ii)). If , (3.1) is trivial.
again establishing (3.1). If , the proof is similar to (3.2).
((ii) ⇒ (i)) Let be a martingale difference sequence in .
Then is an α-strong adapted field with by (ii); we have (2.1) and then is p-uniformly smooth. □
where denotes the indicator function of the set .
When for all , with a note that for all , we have the following corollary.
Then (3.7), (3.8), and (3.9) yield (3.6). □
Remark 3.5 The field of functions with satisfies the property (3.5).
for all and .
4 Application to the strong law of large numbers
By applying theorems in Section 3 we establish some results of strong laws of large numbers for fields of blockwise-α-martingale differences with values in a p-uniformly smooth Banach space.
is p-uniformly smooth.
- (ii)is a blockwise-α-martingale difference field in with respect to the blocks , is a nondecreasing field of positive constants satisfying (4.1). If(4.2)
- (iii)is a blockwise-M-martingale difference field in with respect to the blocks , is a nondecreasing sequence of positive constants satisfying (4.1). If (4.2) holds, then we have
((ii) ⇒ (iii)) When for all , with a note that for all . We have (4.3).
Then by Theorem 2.2 of Hoffmann-Jørgensen and Pisier , is p-uniformly smooth. □
is said to be a strong∗ blockwise-α-martingale difference field if it is a blockwise-α-strong∗ adapted field as well as a blockwise-α-martingale difference field.
is p-uniformly smooth.
- (ii)is a strong∗ blockwise-α-martingale difference field in with respect to the blocks , is a field of positive Borel functions satisfying (3.5). If(4.4)
where , then we have (4.3).
Proof ((i) ⇒ (ii)) By Theorem 3.4 and by the same argument as in the proof of Theorem 4.1.
Then is a blockwise-1-martingale difference and strong∗ adapted field with respect to the blocks in and we have . Put , , , , , and for all . Thus, by (ii) and by the same argument as in the proof of Theorem 4.1, we have (i). □
Proof Using Theorem 4.3 and by the same argument as in the proof of Theorem 4.1, we have (4.5). □
Remark 4.5 In Theorem 4.3, when , , we have the result in Theorem 3.2(ii) in .
The authors would like to express their gratitude to the referees for their detailed comments and valuable suggestions, which helped them to improve the manuscript. The research has been supported by VNU Grant No. QG.13.02.
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