On the Strong Law of Large Numbers for D-dimensional Arrays of Random Variables

In this paper, we provide a necessary and sufficient condition for general d-dimensional arrays of random variables to satisfy strong law of large numbers. Then, we apply the result to obtain some strong laws of large numbers for d-dimensional arrays of blockwise independent and blockwise orthogonal random variables.

+ } of random variables defined on a probability space (Ω, F, P ).Let S n = i≺n X i , and let {α i , 1 i d} be positive constants.In Section 2, we provide a necessary and sufficient condition for to hold.This condition springs from a recent result of Chobanyan, Levental and Mandrekar [1] which provided a condition for strong law of large numbers (SLLN) in the case d = 1 (see Chobanyan, Levental and Mandrekar [1,Theorem 3.3]).Some applications to SLLN for d-dimensional arrays of blockwise independent and blockwise orthogonal random variables are made in Section 3.

Result
We can now state our main result. and Then Assume that (2.1) holds.Since we have that This implies The conclusion (2.1) follows immediately from (2.2), (2.7) and (2.8).

Applications
In this section, we present some applications of Theorem 2.1.A d-dimensional array of random variables {X n , n ∈ Z d + } is said to be blockwise independent (resp., blockwise orthogonal) if for each k ∈ Z d + , the random variables {X i , i ∈ I(k)} is independent (resp., orthogonal).The concept of blockwise independence for a sequence of random variables was introduced by Móricz [9].Extensions of classical Kolmogorov SLLN (see, e.g., Chow and Teicher [2], p. 124) to the blockwise independence case were established by Móricz [9] and Gaposhkin [4].Móricz [9] and Gaposhkin [4] also studied SLLN problem for sequence of blockwise orthogonal random variables.Firstly, we establish a blockwise independence and d-dimensional version of the Kolmogorov SLLN.THEOREM 3.1.Let {X n , n ∈ Z d + } be a d-dimensional array of mean 0 blockwise independent random variables and let {α i , 1 i d} be positive constants.If obtains.
In the case 0 < p 1, the independence hypothesis and the hypothesis that EX n = 0, n ∈ Z d + are superfluous.
Proof.We need the following lemma which was proved by Thanh [11] in the case d = 2.If d is arbitrary positive integer, then the proof is similar and so is omitted.LEMMA 3.1.Let n ∈ Z d + and let {X i , i ≺ n} be a collection of |n| mean 0 independent random variables.Then there exists a constant C depending only on p and d such that In the case 0 < p 1, the independence hypothesis and the hypothesis that EX i = 0, i ≺ n are superfluous, and C is given by C = 1.In the case 1 < p < 2, C is given by C = 2 p p − 1 pd .
In the case p = 2, Lemma 3.1 was proved by Wichura [12] and C is given by C The following theorem extends Theorem 3.1 and its part (ii) reduces to a result of Smythe [10] when the {X n , n ∈ Z d + } are independent and + } be a d-dimensional array of random variables and let {α i , 1 i d} be positive constants.Assume that ϕ(x) is a continuous functions on [0, ∞), ϕ(0) 0, ϕ(x) > 0 for x > 0, and + } are blockwise independent and have mean 0, and Consider the case (i) first.It follows from (3.3) that (by the first condition of (i)) < ∞.
On the other hand (by the second condition of (i)) By the Borel-Cantelli lemma, The conclusion (3.2) follows immediately from (3.4) and (3.5).Now, consider the case (ii).It follows from (3.3) that and The conclusion (3.2) follows immediately from (3.8) and (3.9).REMARK 3.1.(i) According to the discussion in Smythe [10], the proof of part (ii) of Theorem 3.2 was based on the "Khintchin-Kolmogorov convergence theorem, Kronecker lemma approach".But it seems that the Kronecker lemma for d-dimensional arrays when d 2 is not such a good tool as in the study of the SLLN for the case d = 1 (see Mikosch and Norvaisa [6]).Moreover, in the blockwise independence case, according to an example of Móricz [9], the conclusion of Theorem 3.1 (or part (ii) of Theorem 3.2) cannot in general be reached through the well-know Kronecker lemma approach for proving SLLNs even when d = 1.
(ii) Chung [3] proved part (i) of Theorem (3.2) (for the case d = 1 only) by the Kolmogorov three series theorem and the Kronecker lemma.So in his proof, the independence assumption must be required.We now establish the Marcinkiewicz-Zygmund SLLN for d-dimensional arrays of blockwise independent identically distributed random variables.The following theorem reduces to a result of Gut [5] when the {X n , n ∈ Z d + } are independent.THEOREM 3. Proof.According to the proof of Lemma 2.2 of Gut [5], where And similarly, we also have Finally, we establish the SLLN for d-dimensional arrays of blockwise orthogonal random variables.The following theorem is a blockwise orthogonality version of Theorem 1 of Móricz [8] and its proof is based on the d-dimensional version of the Rademacher-Mensov inequality (see Móricz [7]) and the method used in the proof of Theorem 3.1.THEOREM 3.4.Let {X n , n ∈ Z d + } be a d-dimensional array of blockwise orthogonal random variables and let {α i , 1 i d} be positive constants.If
THEOREM 2.1.Let {X n , n ∈ Z d + } be a d-dimensional array of random variables and let {α i , 1 i d} be positive constants.For m = (m 1 , . .., m d ) ∈ Z dProof.To prove Theorem 2.1, we will need the following lemmma.The proof of the following lemma is just an application of Kronecker's lemma with d-dimensional indices as was so kindly pointed out to the author by the referee.LEMMA 2.1.Let {x n , n ∈ Z d + } be a d-dimensional array of constants, and let {α i , 1 i d} be a collection of positive constants.If holds by Lemma 2.1.Thus (2.1) implies (2.2).Now, assume that (2.2)