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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
$S_n=\xi_1+\dots+\xi_n$ 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.
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.
UDC:
519.21
MSC: 60F10 Received August 19, 2009, published
October 8, 2009
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