A Hájek-Rényi-Type Maximal Inequality and Strong Laws of Large Numbers for Multidimensional Arrays

Abstract

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 .

Hájek and Rényi [1] proved the following important inequality: If is a sequence of (real-valued) independent random variables with zero means and finite second moments, and is a nondecreasing sequence of positive real numbers, then for any and for any positive integers ,

(1.1)

This inequality is a generalization of the Kolmogorov inequality and is a useful tool to prove the strong law of large numbers. Fazekas and Klesov [2] gave a general method for obtaining the strong law of large numbers for sequences of random variables by using a Hájek-Rényi-type maximal inequality. Afterwards, Noszály and Tómács [3] extended this result to multidimensional arrays (see also Klesov et al. [4]). They provided a sufficient condition for -dimensional arrays of random variables to satisfy the strong law of large numbers

(1.2)

where is a positive, nondecreasing -sequence of product type, that is, , where is a nondecreasing sequence of positive real numbers for each . Then, we have

(1.3)

This implies that

(1.4)

Therefore,

(1.5)

(1.6)

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).

Thus, if is a positive, nondecreasing -sequence of product type, then it is an array of positive real numbers satisfying (1.5), but the reverse is not true.

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 probability space. A family of nondecreasing sub-σ-algebras of related to the partial order ⪯ on is said to be a stochastic basic.

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.

For a given stochastic basic , for , we set

(1.7)

in the case , we set .

An adapted array is said to be a martingale difference array if for all and for all .

In Quang and Huan [5], 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.

A Banach space is said to be -uniformly smooth () if

(1.8)

A Banach space is said to be -smoothable if there exists an equivalent norm under which is -uniformly smooth.

Pisier [6] proved that a real separable Banach space is -smoothable () if and only if there exists a positive constant such that for every integrable (-valued) martingale difference sequence ,

(1.9)

In Quang and Huan [5], this inequality was used to define -uniformly smooth Banach spaces.

Let be a sequence of independent identically distributed random variables with . Let and define

(1.10)

Let . Then, is said to be of Rademacher type if there exists a positive constant such that

(1.11)

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 [7], Pisier [8], and Woyczyński [9].

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 [10] and is a multidimensional version of the Kronecker lemma.

Lemma 1.1.

Let be an array of nonnegative real numbers, and let be a nondecreasing array of positive real numbers such that as . If

(1.12)

then

(1.13)

Proof.

For every , there exists a point such that

(1.14)

Therefore, for all ,

(1.15)

It means that

(1.16)

On the other hand, since as ,

(1.17)

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.

Lemma 1.2.

Let be a probability space, and let be an array of sets in such that for any points . Then,

(1.18)

Lemma 1.3.

Let be an array of random elements. If for any ,

(1.19)

then .

Proof.

For each , we have

(1.20)

Set

(1.21)

Then, and for all , for any , there exists a point such that for all . It means that

(1.22)

The proof is completed.

Lemma 1.4 (Quang and Huan [5]).

Let , and let be a real separable Banach space. Then, the following two statements are equivalent.

(i)The Banach space is -smoothable.

(ii)For every integrable martingale difference array , there exists a positive constant (depending only on and ) such that

(1.23)

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 [11].

Theorem 2.1.

Let , let be an array of positive real numbers satisfying (1.5), and let be an array of random elements in a real separable Banach space. Then, there exists a positive constant such that for any and for any points ,

(2.1)

Proof.

Since is a nondecreasing array of positive real numbers,

(2.2)

For , set

(2.3)

Then, by interchanging the order of summation, we obtain the following

(2.4)

Thus, since ,

(2.5)

By (2.2) and (2.5) and the Markov inequality, we have

(2.6)

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. [4].

Theorem 2.2.

Let , let be an array of nonnegative real numbers, let be an array of positive real numbers satisfying (1.5) and as , and let be an array of random elements in a real separable Banach space such that for any points ,

(2.7)

Then, the condition

(2.8)

implies (1.2).

Proof.

By (2.7) and Theorem 2.1, for any and for any points , we have

(2.9)

This implies, by letting , that

(2.10)

Letting , by (2.8) and Lemma 1.1, we obtain

(2.11)

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 [5] (stated as Lemma 1.4 above).

Theorem 2.3.

Let , and let be a real separable Banach space. Then, the following four statements are equivalent.

(i)The Banach space is -smoothable.

(ii)For every integrable martingale difference array , there exists a positive constant such that

(2.12)

(iii)For every integrable martingale difference array , 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)

(iv)For every martingale difference array , for every array of positive real numbers satisfying (1.5) and as , the condition

(2.14)

implies (1.2).

Proof.

(i)⇒(ii): We easily obtain (2.12) in the case . Now, we consider the case . By virtue of Lemma 1.4, it suffices to show that

(2.15)

First, we remark that for , (2.15) follows from Doob's inequality. We assume that (2.15) holds for , we wish to show that it holds for .

For , we set

(2.16)

Then,

(2.17)

Therefore,

(2.18)

It means that is a nonnegative submartingale. Applying Doob's inequality, we obtain

(2.19)

We set

(2.20)

Then we again have that is a martingale difference array. Therefore, by the inductive assumption, we obtain

(2.21)

Combining (2.19) and (2.21) yields that (2.15) holds for .

• (ii) ⇒ (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.

• (iii) ⇒ (iv): the proof of this implication is similar to the proof of Theorem 2.2 and is therefore omitted.

• (iv) ⇒ (i): for a given positive integer , assume that (iv) holds. Let be an arbitrary martingale difference sequence such that

  

  (2.22)

For , set

(2.23)

Then, is a martingale difference array, and is an array of positive real numbers satisfying (1.5) and as . Moreover, we see that

(2.24)

and so (1.2) holds. It means that

(2.25)

Then, by Theorem 2.2 of Hoffmann-Jørgensen and Pisier [12], is -smoothable.

Remark 2.4.

The inequality (2.15) holds for every and for every martingale difference array without imposing any geometric condition on the Banach space.

In the case , Theorem 2.3 reduces to the following corollary which was proved by Gan [13] and Gan and Qiu [14].

Corollary 2.5.

Let , and let be a real separable Banach space. Then, the following three statements are equivalent.

(i)The Banach space is -smoothable.

(ii)For every integrable martingale difference sequence , for every nondecreasing sequence of positive real numbers , for any , and for any positive integers , there exists a positive constant such that

(2.26)

(iii)For every martingale difference sequence and for every nondecreasing sequence of positive real numbers such that as , the condition

(2.27)

implies

(2.28)

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 [12] 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 [15] and Gan and Qiu [14]. We omit its proof.

Theorem 2.6.

Let , and let be a real separable Banach space. Then, the following four statements are equivalent.

(i)The Banach space is of Rademacher type .

(ii)For every array of integrable independent mean zero random elements , there exists a positive constant such that (2.12) holds.

(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.

(iv)For every array of independent mean zero random elements , for every array of positive real numbers satisfying (1.5) and as , the condition (2.14) implies (1.2).

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.

Remark 2.7.

By the same method as in the proof of Lemma 3 of Móricz et al. [16] and the same arguments as in the proof of Theorem 2.3, we can extend Theorem 2.6 to -dependent random fields.

Example 2.8.

Let be a positive integer , and let be an array of independent random variables with

(2.29)

Then, is an array of independent mean zero random variables taking values in the 2-smoothable Banach space (using the absolute value as norm).

Let . Then,

(2.30)

It means that is an array of positive real numbers satisfying (1.5) and as . Moreover, by virtue of (1.6), we can show that is not a positive, nondecreasing -sequence of product type. Therefore, (1.2) does not follow from Theorem 3.2 of Klesov et al. [4]. But for every array of positive real numbers , is a martingale difference array such that

(2.31)

and so (2.7) and (2.8) are satisfied, Theorem 2.2 ensures that (1.2) holds.

As we know, the limit is equivalent to . Recently, some authors have derived the sufficient conditions for the strong law of large numbers

(2.32)

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].