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 Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 2, Pages 320–341 (Mi tvp2536)  On probabilities of large deviations for sums of independent random variables

L. V. Osipov

Abstract: Let $X_1,…,X_n,…$ be a sequence of independent identically distributed random variables with distribution function $F(x)$, and let $\mathbf EX_i=0$, $\mathbf DX_i=1$. Put
$$F_n(x)=\mathbf P\{\sum_1^nX_i<x\},\quad\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-z^2/2} dz.$$
Let $\Lambda(z)$ be such a function that $\Lambda(z)/\sqrt z\to\infty$, $z\to\infty$, and $\Lambda(z)<z^\alpha$, $1/2<\alpha<1$. We consider the following problem: under which conditions
$$1-F_n(x)=(1-\Phi(\frac x{\sqrt n}))\exp\{\sum_{\nu=3}^k\mu_\nu\frac{x^\nu}{n^{\nu-1}}\}(1+o(1)),\quad n\to\infty,$$
uniformly in $x\in[0,\Lambda(n)]$ where $k$ is the largest integer for which $\varlimsup_{z\to\infty}\Lambda^k(z)/z^{k-1}>0$ and $\mu_3,…,\mu_k$ are real numbers? Theorem 4 gives an answer to this question under some additional restrictions on $\Lambda(z)$. In Theorem 2 we consider the case $\Lambda(z)=z^\alpha$. Full text: PDF file (990 kB)

English version:
Theory of Probability and its Applications, 1973, 17:2, 309–331 Bibliographic databases:  Citation: L. V. Osipov, “On probabilities of large deviations for sums of independent random variables”, Teor. Veroyatnost. i Primenen., 17:2 (1972), 320–341; Theory Probab. Appl., 17:2 (1973), 309–331 Citation in format AMSBIB
\Bibitem{Osi72} \by L.~V.~Osipov \paper On probabilities of large deviations for sums of independent random variables \jour Teor. Veroyatnost. i Primenen. \yr 1972 \vol 17 \issue 2 \pages 320--341 \mathnet{http://mi.mathnet.ru/tvp2536} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=303583} \zmath{https://zbmath.org/?q=an:0271.60035} \transl \jour Theory Probab. Appl. \yr 1973 \vol 17 \issue 2 \pages 309--331 \crossref{https://doi.org/10.1137/1117034} 

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Erratum

This publication is cited in the following articles:
1. M. S. Ermakov, “On the Lower Bound of the Exact Asymptotics for the Large-Deviation Probabilities of Statistical Estimates”, Problems Inform. Transmission, 35:3 (1999), 236–247   2. A. A. Borovkov, “Large Deviations of Sums of Random Variables of Two Types”, Siberian Adv. Math., 11:4 (2001), 1–24   3. A. A. Borovkov, A. A. Mogul'skii, “Integro-local and integral theorems for sums of random variables with semiexponential distributions”, Siberian Math. J., 47:6 (2006), 990–1026       4. A. A. Borovkov, A. A. Mogul'skii, L. V. Rozovskii, A. I. Sakhanenko, “On Zhulev's paper “On large deviations. II””, Theory Probab. Appl., 51:2 (2007), 398–400      5. M. S. Ermakov, “Nonparametric Hypothesis Testing with Small Type I or Type II Error Probabilities”, Problems Inform. Transmission, 44:2 (2008), 119–137    6. A. A. Mogulskii, “Integralnye i integro-lokalnye teoremy dlya summ sluchainykh velichin s semieksponentsialnymi raspredeleniyami”, Sib. elektron. matem. izv., 6 (2009), 251–271   7. Budhiraja A., Dupuis P., Ganguly A., “Moderate deviation principles for stochastic differential equations with jumps”, Ann. Probab., 44:3 (2016), 1723–1775      8. Lifshits M.A. Nikitin Ya.Yu. Petrov V.V. Zaitsev A.Yu. Zinger A.A., “Toward the History of the Saint Petersburg School of Probability and Statistics. i. Limit Theorems For Sums of Independent Random Variables”, Vestnik St. Petersburg Univ. Math., 51:2 (2018), 144–163  •   Contact us: math-net2020_11 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020