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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.

MSC:
60F15Strong limit theorems
60K05Renewal theory
WorldCat.org
Full Text: DOI
References:
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