A sequence of of random variables is said to be extended negatively dependent if for each n the tails of finite-dimensional distributions of random variables 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 . If such statement holds only for multi-dimensional distributions in the lower-left corner only then the sequence 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 -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 be a sequence of END random variables with common distribution . Let be its nth partial sum, , then as for some real number if and only if and .
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.
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