Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports]
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Sib. Èlektron. Mat. Izv.: Year: Volume: Issue: Page: Find

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

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.

Full text: PDF file (864 kB)
References: PDF file   HTML file

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} `

• http://mi.mathnet.ru/eng/semr67
• http://mi.mathnet.ru/eng/semr/v6/p251

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. A. A. Borovkov, K. A. Borovkov, “Blackwell-type theorems for weighted renewal functions”, Siberian Math. J., 55:4 (2014), 589–605
•  Number of views: This page: 160 Full text: 47 References: 23