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Teor. Veroyatnost. i Primenen., 1962, Volume 7, Issue 4, Pages 433–437 (Mi tvp4739)  

This article is cited in 1 scientific paper (total in 1 paper)

Short Communications

On Convergence in the Mean for Densities

S. Kh. Sirazhdinov, M. Mamatov

V. I. Lenin Tashkent State University

Abstract: A sequence of normed sums $\zeta_n=(\xi _1+\cdots+\xi _n)/\sqrt n$ is considered ( $\xi _1,…,\xi_n$ are equally distributed random variables, $\mathbf M\xi _i=0,\mathbf M\xi_i^2=1$). Let $\varphi (x)$ denote the density of the normal distribution with parameters $(0,1)$, $p_n (x)$ the density of the absolutely continuous component of the distribution of the sum $\zeta _n$. The main results of the paper are as follows: if the condition (A) is satisfied and the components $\xi _i$ have finite third moments $\alpha$, then
$$C_n=\int|p_n(x)-\varphi(x)| dx=\frac{| \alpha|}{\sqrt n}\lambda+o(\frac1{\sqrt n}),$$
where $\lambda$ is a constant, whose value is given in Theorem 1.
The other theorems refer to the case when the moment $\alpha$ does not exist.

Full text: PDF file (488 kB)

English version:
Theory of Probability and its Applications, 1962, 7:4, 424–428

Received: 14.09.1961

Citation: S. Kh. Sirazhdinov, M. Mamatov, “On Convergence in the Mean for Densities”, Teor. Veroyatnost. i Primenen., 7:4 (1962), 433–437; Theory Probab. Appl., 7:4 (1962), 424–428

Citation in format AMSBIB
\Bibitem{SirMam62}
\by S.~Kh.~Sirazhdinov, M.~Mamatov
\paper On Convergence in the Mean for Densities
\jour Teor. Veroyatnost. i Primenen.
\yr 1962
\vol 7
\issue 4
\pages 433--437
\mathnet{http://mi.mathnet.ru/tvp4739}
\transl
\jour Theory Probab. Appl.
\yr 1962
\vol 7
\issue 4
\pages 424--428
\crossref{https://doi.org/10.1137/1107039}


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    This publication is cited in the following articles:
    1. Bally V., Caramellino L., Poly G., “Convergence in Distribution Norms in the Clt For Non Identical Distributed Random Variables”, Electron. J. Probab., 23 (2018), 45  crossref  mathscinet  zmath  isi  scopus
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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