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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1961, Volume 6, Issue 4, Pages 377–391 (Mi tvp4795)

Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations. II

Yu. V. Linnik

Abstract: “Narrow” Zones of Local and Integral Normal Attraction. Using the notation in Part I of this article, we consider the integral normal attraction zones for the variables $X_i$ and local normal attraction zones for $X_j\in(d)$. The monotone function $h(x)\leq x^{1/2}$ is considered under the supplementary conditions explained in Part I; the “narrow zone theorems” are more conveniently expressed in terms of the condition
$$\label{eq*}\tag{*} E\exp h(|X_j |)<\infty.$$
The equation
$$h(\sqrt n\Lambda(n))=(\Lambda(n))^2$$
determines the monotone function $\Lambda (n)$. The condition \eqref{eq*} is necessary for the zones $[0,\Lambda (n)\rho (n)],[ - \Lambda (n)\rho (n),0]$ to be z.n.a., and for $X_j \in (d)$ to be z.u.l.n.a. It is sufficientt for the zones $[0,\Lambda (n)/\rho(n)], [-\Lambda(n)/\rho (n),0]$ to be z.n.a. and for $X_j\in(d)$ – to be z.u.l.n.a.

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English version:
Theory of Probability and its Applications, 1961, 6:4, 345–360

Citation: Yu. V. Linnik, “Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations. II”, Teor. Veroyatnost. i Primenen., 6:4 (1961), 377–391; Theory Probab. Appl., 6:4 (1961), 345–360

Citation in format AMSBIB
\Bibitem{Lin61} \by Yu.~V.~Linnik \paper Limit Theorems for Sums of Independent Variables Taking into Account Large Deviations.~II \jour Teor. Veroyatnost. i Primenen. \yr 1961 \vol 6 \issue 4 \pages 377--391 \mathnet{http://mi.mathnet.ru/tvp4795} \transl \jour Theory Probab. Appl. \yr 1961 \vol 6 \issue 4 \pages 345--360 \crossref{https://doi.org/10.1137/1106048} 

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This publication is cited in the following articles:
1. D. V. Batkovich, “Local limit theorems for large deviations”, J. Math. Sci. (N. Y.), 188:6 (2013), 641–654
2. 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