This article is cited in 3 papers

Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

V. V. Shneer

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Let $\{\xi_i\}_{i=1}$ be a sequence of independent identically distributed nonnegative random variables, $S_n=\xi_1+\dots+\xi_n$. Let $\Delta=(0,T]$ and $x+\Delta=(x,x+T]$. We study the ratios of the probabilities $\mathbf{P}(s_n\in x+\Delta)/\mathbf{P}(\xi\in x+\Delta)$ for all $n$ and $x$. The estimates uniform in $x$ for these ratios are known for the so-called $\Delta$-subexponential distributions. Here we improve these estimates for two subclasses of $\Delta$-subexponential distributions; one of them is a generalization of the well-known class $\mathscr{SC}$ to the case of the interval $(0,T]$ with an arbitrary $T\leqslant\infty$. Also, a characterization of the class $\mathscr{SC}$ is given.

Keywords: subexponential distribution, locally subexponential distribution, sums of random variables, estimates for interval probabilities.

UDC: 519.21

Received: 26.05.2005
Revised: 14.04.2006

 English version: Siberian Mathematical Journal, 2006, 47:4, 779–786

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