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 Teor. Veroyatnost. i Primenen., 2003, Volume 48, Issue 1, Pages 78–103 (Mi tvp302)

Superlarge deviations of a sum of independent random variables having a common absolutely continuous distribution under the Cramér condition

L. V. Rozovskii

Abstract: This paper studies the asymptotic behavior of a density of a sum of independent identically distributed random variables with a common absolutely continuous distribution satisfying the right-hand Cramér condition. We prove that for a definite class of such distributions the well-known asymptotic representations in local and integral limit theorems are valid in the case of large deviations of arbitrarily high order.

Keywords: independent random variables, density function, large deviations, Cramér condition.

DOI: https://doi.org/10.4213/tvp302

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English version:
Theory of Probability and its Applications, 2004, 48:1, 108–130

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Citation: L. V. Rozovskii, “Superlarge deviations of a sum of independent random variables having a common absolutely continuous distribution under the Cramér condition”, Teor. Veroyatnost. i Primenen., 48:1 (2003), 78–103; Theory Probab. Appl., 48:1 (2004), 108–130

Citation in format AMSBIB
\Bibitem{Roz03} \by L.~V.~Rozovskii \paper Superlarge deviations of a sum of independent random variables having a common absolutely continuous distribution under the Cram\'er condition \jour Teor. Veroyatnost. i Primenen. \yr 2003 \vol 48 \issue 1 \pages 78--103 \mathnet{http://mi.mathnet.ru/tvp302} \crossref{https://doi.org/10.4213/tvp302} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2013406} \zmath{https://zbmath.org/?q=an:1056.60023} \transl \jour Theory Probab. Appl. \yr 2004 \vol 48 \issue 1 \pages 108--130 \crossref{https://doi.org/10.1137/S0040585X980233} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000220694300007} 

• http://mi.mathnet.ru/eng/tvp302
• https://doi.org/10.4213/tvp302
• http://mi.mathnet.ru/eng/tvp/v48/i1/p78

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This publication is cited in the following articles:
1. A. A. Borovkov, A. A. Mogul'skii, “On large and superlarge deviations for sums of independent random vectors under the Cramer condition. I”, Theory Probab. Appl., 51:2 (2007), 227–255
2. L. V. Rozovskii, “Superlarge deviation probabilities for sums of independent random variables with exponential decreasing distribution”, Theory Probab. Appl., 52:1 (2008), 167–171
3. L. V. Rozovsky, “Small deviation probabilities for sums of independent positive random variables, whose density has a power decay at zero”, J. Math. Sci. (N. Y.), 188:6 (2013), 748–752
4. L. V. Rozovskii, “Superlarge deviation probabilities for sums of independent random variables with exponential decreasing distributions. II”, Theory Probab. Appl., 59:1 (2015), 168–177
5. L. V. Rozovskii, “Ob asimptotike svertki raspredelenii s regulyarno eksponentsialno ubyvayuschimi khvostami”, Veroyatnost i statistika. 28, Zap. nauchn. sem. POMI, 486, POMI, SPb., 2019, 265–274
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