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

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

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

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