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Strong limit theorems for weighted sums of NOD sequence and exponential inequalities. (English) Zbl 1234.60035
Many of classical limit theorems for the sum of independent identically distributed (i.i.d.) random variables have been extended to weighted sums of i.i.d. random variables. This paper considers the extensions of some strong limit theorems to the weighted sum of a sequence of negatively orthant-dependent (NOD) random variables introduced by K. Joag-Dev and F. Proschan [Ann. Stat. 11, 286–295 (1983; Zbl 0508.62041)]. After establishing some basic properties of NOD random variables, the authors provide some results on the complete and almost surely convergences of the weighted sum of NOD sequences, which are generalizations of results provided by R. Giuliano Antonini, J. S. Kwon, S. H. Sung and A. I. Volodin [Stochastic Anal. Appl. 19, No. 6, 903–909 (2001; Zbl 0989.60037)] for the weighted sum of i.i.d. random variables. Finally, an exponential inequality is obtained for bounded NOD sequences, which may be useful to obtain the rate of convergence for NOD sequences, although no result in this direction is given in this paper.

MSC:
60F15Strong limit theorems
60E15Inequalities in probability theory; stochastic orderings