A Hájek-Rényi-Type Maximal Inequality and Strong Laws of Large Numbers for Multidimensional Arrays
© Nguyen Van Quang and Nguyen Van Huan. 2010
Received: 1 July 2010
Accepted: 27 October 2010
Published: 31 October 2010
A Hájek-Rényi-type maximal inequality is established for multidimensional arrays of random elements. Using this result, we establish some strong laws of large numbers for multidimensional arrays. We also provide some characterizations of Banach spaces.
1. Introduction and Preliminaries
Throughout this paper, the symbol will denote a generic positive constant which is not necessarily the same one in each appearance. Let be a positive integer, the set of all nonnegative integer -dimensional lattice points will be denoted by , and the set of all positive integer -dimensional lattice points will be denoted by . We will write , , , and for points , , , and , respectively. The notation (or ) means that for all , the limit is interpreted as for all (this limit is equivalent to ), and we define .
Let be a -dimensional array of real numbers. We define to be the th-order finite difference of the 's at the point . Thus, for all . For example, if , then for all , (with the convention that ). We say that is a nondecreasing array if for any points .
On the other hand, we can show that under the assumption that is an array of positive real numbers satisfying (1.5), it is not possible to guarantee that (1.6) holds (for details, see Example 2.8 in the next section).
In this paper, we use the hypothesis that is an array of positive real numbers satisfying (1.5) and continue to study the problem of finding the sufficient condition for the strong law of large numbers (1.2). We also establish a Hájek-Rényi-type maximal inequality for multidimensional arrays of random elements and some maximal moment inequalities for arrays of dependent random elements.
The paper is organized as follows. In the rest of this section, we recall some definitions and present some lemmas. Section 2 is devoted to our main results and their proofs.
Let be a stochastic basic such that if , let be a real separable Banach space, let be the -algebra of all Borel sets in , and let be an array of random elements such that is -measurable for all . Then is said to be an adapted array.
In Quang and Huan , the authors showed that the set of all martingale difference arrays is really larger than the set of all arrays of independent mean zero random elements.
In Quang and Huan , this inequality was used to define -uniformly smooth Banach spaces.
It is well known that if a real separable Banach space is of Rademacher type , then it is of Rademacher type for all . Every real separable Banach space is of Rademacher type 1, while the -spaces and -spaces are of Rademacher type for . The real line is of Rademacher type 2. Furthermore, if a Banach space is -smoothable, then it is of Rademacher type . For more details, the reader may refer to Borovskikh and Korolyuk , Pisier , and Woyczyński .
Now, we present some lemmas which will be needed in what follows. The first lemma is a variation of Lemma 2.6 of Fazekas and Tómács  and is a multidimensional version of the Kronecker lemma.
Combining the above arguments, this completes the proof of Lemma 1.1.
The proof of the next lemma is very simple and is therefore omitted.
The proof is completed.
Lemma 1.4 (Quang and Huan ).
2. Main Results
Theorem 2.1 provides a Hájek-Rényi-type maximal inequality for multidimensional arrays of random elements. This theorem is inspired by the work of Shorack and Smythe .
This completes the proof of the theorem.
Now, we use Theorem 2.1 to prove a strong law of large numbers for multidimensional arrays of random elements. This result is inspired by Theorem 3.2 of Klesov et al. .
Lemma 1.3 ensures that (1.2) holds. The proof is completed.
The next theorem provides three characterizations of -smoothable Banach spaces. The equivalence of (i) and (ii) is an improvement of a result of Quang and Huan  (stated as Lemma 1.4 above).
⇒ (iii): let be an arbitrary integrable martingale difference array. Then, for all is also an integrable martingale difference array. Therefore, the assertion (ii) and Theorem 2.1 ensure that (2.13) holds.
⇒ (iv): the proof of this implication is similar to the proof of Theorem 2.2 and is therefore omitted.
Then, by Theorem 2.2 of Hoffmann-Jørgensen and Pisier , is -smoothable.
Remark 2.4 ensures that the inequality (2.15) holds for every and for every array of independent mean zero random elements in a real separable Banach space. Therefore, by using the implication ((2.1.1) ⇒ (2.1.2)) of Theorem 2.1 of Hoffmann-Jørgensen and Pisier  and the same arguments as in the proof of Theorem 2.3, we get the following theorem which generalizes some results given by Christofides and Serfling  and Gan and Qiu . We omit its proof.
(iii)For every array of integrable independent mean zero random elements , for every array of positive real numbers satisfying (1.5), for any , and for any points , there exists a positive constant such that (2.13) holds.
We close this paper by giving a remark on Theorem 2.6 and an example which illustrates Theorems 2.2, 2.3, and 2.6.
By the same method as in the proof of Lemma 3 of Móricz et al.  and the same arguments as in the proof of Theorem 2.3, we can extend Theorem 2.6 to -dependent random fields.
and so (2.7) and (2.8) are satisfied, Theorem 2.2 ensures that (1.2) holds.
where is one of the special kinds of positive, nondecreasing -sequences of product type. For more details, the reader may refer to [17–19]. Therefore, this example also shows that the implications ((i) ⇒ (iv)) of Theorem 2.3 and ((i) ⇒ (iv)) of Theorem 2.6 are independent of results obtained in [17–19].
The authors are grateful to the referee for carefully reading the paper and for offering some comments which helped to improve the paper. This research was supported by the National Foundation for Science Technology Development, Vietnam (NAFOSTED), no. 101.02.32.09.
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