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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 integro-local 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 integro-local 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, 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.

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Bibliographic databases:
UDC: 519.21
MSC: 60F10
Received August 19, 2009, published October 8, 2009

Citation: A. A. Mogul'skii, “Integral and integro-local 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 integro-local theorems for the sums of random variables with semiexponential distribution
\jour Sib. \`Elektron. Mat. Izv.
\yr 2009
\vol 6
\pages 251--271
\mathnet{http://mi.mathnet.ru/semr67}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2586690}
\elib{https://elibrary.ru/item.asp?id=13035587}


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    This publication is cited in the following articles:
    1. Mikosch T., Wintenberger O., “Precise Large Deviations for Dependent Regularly Varying Sequences”, Probab. Theory Relat. Field, 156:3-4 (2013), 851–887  crossref  mathscinet  zmath  isi
    2. A. A. Borovkov, K. A. Borovkov, “Blackwell-type theorems for weighted renewal functions”, Siberian Math. J., 55:4 (2014), 589–605  mathnet  crossref  mathscinet  isi
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