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 Mat. Tr., 2008, Volume 11, Number 1, Pages 81–112 (Mi mt118)

Superlarge deviations for sums of random variables with arithmetical super-exponential distributions

A. A. Mogulskiĭab, Ch. Pagma

a Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences
b Novosibirsk State University

Abstract: Local limit theorems are obtained for superlarge deviations of sums $S(n)=\xi(1)+…+\xi(n)$ of independent identically distributed random variables having an arithmetical distribution with the right-hand tail decreasing faster that that of a Gaussian law. The distribution of $\xi$ has the form $\mathbb P(\xi=k)=e^{-k^\beta L(k)}$, where $\beta>2$, $k\in\mathbb Z$ ($\mathbb Z$ is the set of all integers), and $L(t)$ is a slowly varying function as $t\to\infty$ which satisfies some regularity conditions. These theorems describing an asymptotic behavior of the probabilities $\mathbb P(S(n)=k)$ as $k/n\to\infty$, complement the results on superlarge deviations in [1, 2].

Key words: arithmetical super-exponential distribution, integro-local and local theorems, superlarge deviations, deviation function, random walk, Gaussian approximation, Poissonian approximation.

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English version:
Siberian Advances in Mathematics, 2008, 18:3, 185–208

Bibliographic databases:

UDC: 514.76+517.98

Citation: A. A. Mogulskiǐ, Ch. Pagma, “Superlarge deviations for sums of random variables with arithmetical super-exponential distributions”, Mat. Tr., 11:1 (2008), 81–112; Siberian Adv. Math., 18:3 (2008), 185–208

Citation in format AMSBIB
\Bibitem{MogPag08} \by A.~A.~Mogulski{\v\i}, Ch.~Pagma \paper Superlarge deviations for sums of random variables with arithmetical super-exponential distributions \jour Mat. Tr. \yr 2008 \vol 11 \issue 1 \pages 81--112 \mathnet{http://mi.mathnet.ru/mt118} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2437483} \transl \jour Siberian Adv. Math. \yr 2008 \vol 18 \issue 3 \pages 185--208 \crossref{https://doi.org/10.3103/S1055134408030048}