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 Teor. Veroyatnost. i Primenen., 1964, Volume 9, Issue 2, Pages 343–352 (Mi tvp380)

Short Communications

On Local Limit Theorems for the Sums of Independent Random Variables

V. V. Petrov

Abstract: Let $X_1,X_2,…$ be a sequence of independent identically distributed random variables, ${\mathbf E}X_1=m$, ${\mathbf D}X_1=\sigma^2>0$, and ${\mathbf E}|X_1|^k<\infty$ for some integer $k\geqq 3$. The following theorem is proved:
Suppose that the variable $Z_n=\dfrac1{\sigma\sqrt n}(\sum\limits_{j=1}^n{X_j-nm})$ has an absolutely continuous distribution with bounded density function $p_n(x)$ for some integer $n=n_0$. Then there exists a function $\varepsilon(n)$ such that lim $\varepsilon(n)=0$ and relation (1) is fulfilled.
A similar theorem is proved for the case when $X_1$ has a lattice distribution. Some consequences of these theorems concerning convergence to the normal law in the mean are discussed.

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English version:
Theory of Probability and its Applications, 1964, 9:2, 312–320

Bibliographic databases:

Citation: V. V. Petrov, “On Local Limit Theorems for the Sums of Independent Random Variables”, Teor. Veroyatnost. i Primenen., 9:2 (1964), 343–352; Theory Probab. Appl., 9:2 (1964), 312–320

Citation in format AMSBIB
\Bibitem{Pet64} \by V.~V.~Petrov \paper On Local Limit Theorems for the Sums of Independent Random Variables \jour Teor. Veroyatnost. i Primenen. \yr 1964 \vol 9 \issue 2 \pages 343--352 \mathnet{http://mi.mathnet.ru/tvp380} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=163341} \zmath{https://zbmath.org/?q=an:0146.38003} \transl \jour Theory Probab. Appl. \yr 1964 \vol 9 \issue 2 \pages 312--320 \crossref{https://doi.org/10.1137/1109044} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bobkov S.G., “Asymptotic Expansions For Products of Characteristic Functions Under Moment Assumptions of Non-Integer Orders”, Convexity and Concentration, IMA Volumes in Mathematics and Its Applications, 161, ed. Carlen E. Madiman M. Werner E., Springer, 2017, 297–357
2. M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163
3. S. G. Bobkov, A. Marsiglietti, “Local limit theorems for smoothed Bernoulli and other convolutions”, Teoriya veroyatn. i ee primen., 65:1 (2020), 79–102
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