## 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:

 60F15 Strong limit theorems 60K05 Renewal theory
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### References:

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