On Convergence in the Mean for Densities

S. Kh. Sirazhdinov ; M. Mamatov

Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 4, pp. 433-437

Résumé

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.