Unimprovable exponential bounds for distributions of sums of a~random number of random variables

A. A. Borovkov

Teoriâ veroâtnostej i ee primeneniâ, Tome 40 (1995) no. 2, pp. 260-269

Résumé

The basic object of the study is the asymptotic behavior of $\mathbf{P}(Z_\nu>t)$ as $t\to\infty $ for sums $Z_\nu$ of random number $\nu$ of random variables $\zeta_1,\zeta_2,\dots$ . It was established in [1] that, if conditional “with respect to the past” probabilities of the events $\{\zeta_k>t\}$ are dominated by a function $\delta_1(t)$, $\mathbf{P}(\nu>t)\delta_2(t)$, and the functions $\delta_1$ and $\delta_2$ are close to power functions, then $\mathbf{P}(Z_\nu>t)$, $c=\mathrm{const}$, and this bound cannot be improved. In the present paper, we study the asymptotics of $\mathbf{P}(Z_\nu>t)$ in the case when the functions $\delta_1$ and $\delta_2$ are exponential. The nature of unimprovable bounds for $\mathbf{P}(Z_\nu>t)$ turns out in this case to be different.

Détail

Keywords: sums of random number of random variables, stopped sums, large deviations, exponential bounds.