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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

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