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The strong law of large numbers for extended negatively dependent random variables. (English) Zbl 1213.60058

A sequence of {X k ,k=1,2,} of random variables is said to be extended negatively dependent if for each n the tails of finite-dimensional distributions of random variables {X k ,k=1,2,,n} in the lower-left and the upper-right corners are dominated by multiple of tails of corresponding distributions of a sequence of independent random variables with the same marginal distributions as have the random variables X k . If such statement holds only for multi-dimensional distributions in the lower-left corner only then the sequence {X k ,k=1,2,} is called lower extended negative dependent (LEND), if it holds for the upper-right corner only, then we deal with the upper extended negative dependent (UEND) sequence. Sufficient conditions for LEND or for UEND are given in Lemma 2.1. By this lemma every n-dimensional Farlier-Gumber-Morgenstern distribution describes a specifies END structure.

The references on investigations in cases of various negative dependences are given in the Introduction.

The main statement is Theorem 1.1. Let {X k ,k=1,2,} be a sequence of END random variables with common distribution F. Let S n be its nth partial sum, n=1,2,, then S n nμ as n for some real number μ if and only if E|X 1 |< and μ=EX 1 .

In Section 2 six lemmas are presented five of which need proving and one of the is new even for the independent case. Theorem 1.1. is proved in section 3. Section 4 contains two applications of Theorem 1.1. to risk theory and renewal theory.

The volume of the paper is 15 pages. The list of references contains 25 positions.


MSC:
60F15Strong limit theorems
60K05Renewal theory