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Teor. Veroyatnost. i Primenen., 1972, Volume 17, Issue 2, Pages 320–341 (Mi tvp2536)  

This article is cited in 7 scientific papers (total in 8 papers)

On probabilities of large deviations for sums of independent random variables

L. V. Osipov

Leningrad

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 $\Lambda(z)=z^\alpha$.

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English version:
Theory of Probability and its Applications, 1973, 17:2, 309–331

Bibliographic databases:

Received: 24.09.1970

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  mathnet  mathscinet  zmath
    2. A. A. Borovkov, “Large Deviations of Sums of Random Variables of Two Types”, Siberian Adv. Math., 11:4 (2001), 1–24  mathnet  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  elib
    5. M. S. Ermakov, “Nonparametric Hypothesis Testing with Small Type I or Type II Error Probabilities”, Problems Inform. Transmission, 44:2 (2008), 119–137  mathnet  crossref  mathscinet  isi
    6. A. A. Mogulskii, “Integralnye i integro-lokalnye teoremy dlya summ sluchainykh velichin s semieksponentsialnymi raspredeleniyami”, Sib. elektron. matem. izv., 6 (2009), 251–271  mathnet  mathscinet  elib
    7. Budhiraja A., Dupuis P., Ganguly A., “Moderate deviation principles for stochastic differential equations with jumps”, Ann. Probab., 44:3 (2016), 1723–1775  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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