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

A sequence of $\left\{{X}_{k},k=1,2,\cdots \right\}$ of random variables is said to be extended negatively dependent if for each n the tails of finite-dimensional distributions of random variables $\left\{{X}_{k},k=1,2,\cdots ,n\right\}$ 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 $\left\{{X}_{k},k=1,2,\cdots \right\}$ 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 $\left\{{X}_{k},k=1,2,\cdots \right\}$ be a sequence of END random variables with common distribution $F$. Let ${S}_{n}$ be its nth partial sum, $n=1,2,\cdots$, then $\frac{{S}_{n}}{n}\to \mu$ as $n\to \infty$ for some real number $\mu$ if and only if $E|{X}_{1}|<\infty$ and $\mu =E{X}_{1}$.

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:
 60F15 Strong limit theorems 60K05 Renewal theory