MARCINKIEWICZ-TYPE STRONG LAW OF LARGE NUMBERS FOR DOUBLE ARRAYS OF PAIRWISE INDEPENDENT RANDOM VARIABLES

Let { Xij } be a double sequence of pairwise independent random variables. If P {|Xmn| ≥ t}≤ P{|X| ≥ t} for all nonnegative real numbers t and E|X|p( log+ |X|)3 <∞, for 1<p < 2, then we prove that ∑m i=1 ∑n j=1 ( Xij−EXij ) (mn)1/p → 0 a.s. as m∨n →∞. (0.1) Under the weak condition of E|X|p log+ |X| <∞, it converges to 0 in L1. And the results can be generalized to an r -dimensional array of random variables under the conditions E|X|p( log+ |X|)r+1 <∞, E|X|p( log+ |X|)r−1 <∞, respectively, thus, extending Choi and Sung’s result [1] of the one-dimensional case.

1. Introduction.Etemadi [3] extended the classical law of large numbers for i.i.d.random variables to the case where the random variables are pairwise i.i.d., i.e., if {X n } is a sequence of pairwise i.i.d.random variables with E|X 1 | < ∞, then n i=1 X i − EX i n → 0 a.s. (1.1) In 1985, Choi and Sung [1] have shown that if {X n } are pairwise independent and are dominated in distribution by a random variable X with E|X| p log + |X| 2 → 0 a.s.In addition, if E|X| p < ∞, then it converges to 0 in L 1 .
For a double sequence X ij of pairwise i.i.d.random variables, also Etemadi [3] Now, we are interested in the extension of Choi and Sung's result of the onedimensional case to a multi-dimensional array of pairwise independent random variables, which is established in the next section.

Main results.
Let X ij be a double sequence of random variables and let Throughout this paper, c denotes an unimportant positive constant which is allowed to change and d k the number of all divisors of integer k.
To prove the main theorem, we need the following lemmas.
Lemma 2.1.Let X ij be a double sequence of pairwise independent random variables.If P |X mn | ≥ t ≤ P |X| ≥ t for all nonnegative real numbers t, then (2.1) where F(x) is the distribution of X.If we use the fact that where we use the fact that (2.5) By the fact that log n , we can obtain (b) by the same method.
The following lemma is a two parameter analog of [5,Lem. 3.6.1a].
Lemma 2.2.Let X ij be a double sequence of pairwise independent random variables with EX ij = 0, and let Proof.For m = 1 and n = 1, the inequality is trivial.If m > 1, let s be an integer such that 2 s−1 < m ≤ 2 s .And if n > 1, let t be an integer such that 2 t−1 < n ≤ 2 t .We can assume that m, n > 1.We assign X ij to the point (i, j) of integer in (0, , each of these two intervals into two halves, and so on.Then the elements of the ith partition are of length 2 s−i , i = 0,...,s.Also, divide the interval (0, 2 t ] in the same way.Then we obtain the (i, j)th partition P ij of (0, 2 s ] × (0, 2 t ] by the ith partition of (0, 2 s ] and the jth partition of (0, 2 t ].Every rectangle (0,i] × (0,j] is the sum of at most (s + 1)(t + 1) disjoint subrectangles each of which belongs to a different partition.We can write S ij = ( Hence, by the Borel-Cantelli lemma, Now, we use Chebyshev's inequality and Lemma 2.1 to obtain which follows easily by summation by parts.It follows that

.13)
And let where S * mn = S mn − ES mn .By using Lemma 2.2, we obtain, for any > 0, where the last inequality follows easily be summation by parts.But where we use ;ij , where Y kl;ij is the sum of all r.v.'s belonging to the rectangle (a, b] × (c, d], b − a = 2 k and d − c = 2 l , which may or may not be a summand of (0,i] × (0,j] so that some Y kl;ij may vanish.Let T ij = 2 i k=1 2 j l=1 |Y kl | 2 , where Y kl is the sum of all r.v.'s which belong to the (k, l)-element of P ij .If we put T = s i=0 t j=0 T ij , by the elementary Schwarz inequality, we obtain Let X ij be a double sequence of pairwise independent random variables.If P |X mn | ≥ t ≤ P |X| ≥ t for all nonnegative real numbers t and E|X| p log + |X| 3< ∞, for 1 < p < 2, then The generalization to r -dimensional arrays of random variables can be obtained easily under the condition E|X| p log + |X| Corollary 2.4.Let X ij be a double sequence of pairwise i.i.d.random variables with E|X 11 | p log + |X 11 | 3 < ∞, for 1 < p < 2. Then r +1 < ∞.Theorem 2.5.Let X ij be a double sequence of pairwise independent random variables.If P |X ij | ≥ t ≤ P |X| ≥ t for all nonnegative real numbers t and E|X| /(mn) 2/p converges to 0 as m ∨ n → 0. But this can be shown by a method similar to that used in the proof of(2.23)inTheorem 2.3.Let X ij be a double sequence of pairwise i.i.d.random variable with E|X 11 | p log + |X 11 | < ∞, for 1 < p < 2. Then The generalization to r -dimensional arrays of random variables can be obtained under the condition E|X| p log + |X| r +1 < ∞.