Marcinkiewicz-type strong law of large numbers for double arrays of pairwise independent random variables. (1st March 1999)

• Record Type: Journal Article
• Authors: Hong, Dug Hun; Hwang, Seok Yoon
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 22:Number 1(1999)
• Page Start: 171
• Page End: 177
• Publication Date: 1999-03-01
• Subjects: Strong law of large numbers -- pairwise independent; Mathematics -- Periodicals; 510.5
• DOI: 10.1155/S0161171299221710
• Languages: English
• ISSNs: 0161-1712

Abstract

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 andE | X | p ( log + | X | ) 3 < ∞, for1 < 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 ofE | X | p log + | X | < ∞, it converges to 0 inL 1 . And the results can be generalized to anr -dimensional array of random variables under the conditionsE | 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.