Abstract
In this paper, necessary and sufficient conditions of the complete convergence are obtained for the maximum partial sums of negatively orthant dependent (NOD) random variables. The results extend and improve those in Kuczmaszewska (Acta Math. Hung. 128(1-2):116-130, 2010) for negatively associated (NA) random variables.MSC:60F15, 60G50.
Highlights
1 Introduction The concept of complete convergence for a sequence of random variables was introduced by Hsu and Robbins [ ] as follows
Theorem A Let {Xn, n ≥ } be a sequence of negatively associated (NA) random variables and X be a random variables possibly defined on a different space satisfying the condition
The aim of this paper is to extend and improve Theorem A to negatively orthant dependent (NOD) random variables
Summary
The concept of complete convergence for a sequence of random variables was introduced by Hsu and Robbins [ ] as follows. A sequence {Un, n ≥ } of random variables converges completely to the constant θ if. They proved that the sequence of arithmetic means of independent identically distribution (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite This result has been generalized and extended in several directions by many authors. (Asadian et al [ ]) For any q ≥ , there is a positive constant C(q) depending only on q such that if {Xn, n ≥ } is a sequence of NOD random variables with EXn = for every n ≥ , for all n ≥ , n q. For any q ≥ , there is a positive constant C(q) depending only on q such that if {Xn, n ≥ } is a sequence of NOD random variables with EXn = for every n ≥ , for all n ≥ , j. By Lemma . for ∀n ≥ , {Xi(n, ) – EXi(n, ), ≤ i ≤ n} is a sequence of NOD random variables
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