Integral and integro-local theorems for the sums of random variables with semiexponential distribution

A. A. Mogul'skii

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 6 (2009), pp. 251-271

Résumé

In the present paper, as in [1], we obtain some integral and integro-local theorems for the sums $S_n=\xi_1+\dots+\xi_n$ of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form $\mathbf P(\xi\ge t)=e^{-t^\beta L(t)}$, where $\beta\in(0,1)$ and $L(t)$ is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as $x\to\infty$ of the probabilities $$ \mathbf P(S_n\ge x)\quad\text{and}\quad\mathbf P(S_n\in[x,x+\Delta)) $$ on the whole semiaxis (i.e., in the zone of normal deviations and all zones of large deviations of $x$: in the Cramér and intermediate zones, and also in the “extrem” zone where the distribution of $S_n$ is approximated by that of maximal summand). In the present paper (in contrast to [1]) we have used the minimal moment condition $\mathbf E\xi^2\infty$ on the left tail of the distribution. Under this condition we can not define a segment of the Cramér series (the probabilities under consideration were described via the segment of the Cramér series in the Cramér and intermediate zones in [1]), and have to consider another characteristic instead of it.

Détail

Keywords: semiexponential distribution, deviation function, integral theorem, integro-local theorem, segment of Cramér series, random walk, large deviations, Cramér zone of deviations, intermediate zone of deviations, zone of approximated by the maximal summand.