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

This article is cited in 3 scientific papers (total in 3 papers)

Short Communications

On Local Limit Theorems for the Sums of Independent Random Variables

V. V. Petrov

Leningrad

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.

Full text: PDF file (461 kB)

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

Bibliographic databases:

Received: 15.05.1962

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
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    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  crossref  isi
    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  crossref  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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