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The strong law of large numbers for extended negatively dependent random variables. (English) Zbl 1213.60058
A sequence of $\{X_k, k=1,2,\dots\}$ of random variables is said to be extended negatively dependent if for each n the tails of finite-dimensional distributions of random variables $\{X_k, k=1,2,\dots,n\}$ 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 $\{X_k, k=1,2,\dots\}$ 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 $\{X_k, k=1,2,\dots\}$ be a sequence of END random variables with common distribution $F$. Let $S_n$ be its nth partial sum, $n=1,2,\dots$, then $\frac{S_n}{n}\rightarrow \mu$ as $n\rightarrow \infty$ for some real number $\mu$ if and only if $E|X_1|<\infty$ and $\mu = EX_1$. 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.

60F15Strong limit theorems
60K05Renewal theory
Full Text: DOI
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