
Sib. Èlektron. Mat. Izv., 2009, Volume 6, Pages 251–271
(Mi semr67)




This article is cited in 2 scientific papers (total in 2 papers)
Research papers
Integral and integrolocal theorems for the sums of random variables with semiexponential distribution
A. A. Mogul'skii^{} ^{} Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
Abstract:
In the present paper, as in [1], we obtain some integral and integrolocal theorems for the sums
$S_n=\xi_1+…+\xi_n$ 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
$$
\mathbf P(S_n\ge x)\quadand\quad\mathbf P(S_n\in[x,x+\Delta))
$$
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.
Keywords:
semiexponential distribution, deviation function, integral theorem, integrolocal theorem, segment of Cramér series, random walk, large deviations, Cramér zone of deviations, intermediate zone of deviations, zone of approximated by the maximal summand.
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519.21
MSC: 60F10 Received August 19, 2009, published October 8, 2009
Citation:
A. A. Mogul'skii, “Integral and integrolocal theorems for the sums of random variables with semiexponential distribution”, Sib. Èlektron. Mat. Izv., 6 (2009), 251–271
Citation in format AMSBIB
\Bibitem{Mog09}
\by A.~A.~Mogul'skii
\paper Integral and integrolocal theorems for the sums of random variables with semiexponential distribution
\jour Sib. \`Elektron. Mat. Izv.
\yr 2009
\vol 6
\pages 251271
\mathnet{http://mi.mathnet.ru/semr67}
\mathscinet{http://www.ams.org/mathscinetgetitem?mr=2586690}
\elib{https://elibrary.ru/item.asp?id=13035587}
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