Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables

Dug Hun Hong and Seok Yoon Hwang

International Journal of Mathematics and Mathematical Sciences, 1999, vol. 22, 1-7

Abstract:

Let { X i j } be a double sequence of pairwise independent random variables. If P { | X m n | ≥ 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 ∑ i = 1 m ∑ j = 1 n ( X i j − E X i j ) ( m n ) 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 L 1 . 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.

Date: 1999
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:690634

DOI: 10.1155/S0161171299221710

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