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Teoriya Veroyatnostei i ee Primeneniya, 1962, Volume 7, Issue 4, Pages 433–437
(Mi tvp4739)
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This article is cited in 19 scientific papers (total in 19 papers)
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 ( $\xi _1,\dots,\xi_n$ 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 $$C_n=\int|p_n(x)-\varphi(x)|\,dx=\frac{| \alpha|}{\sqrt n}\lambda+o\left(\frac1{\sqrt n}\right),$$ 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.
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
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