On the accuracy of Gaussian approximation to the distribution functions of sums of independent random variables

I. A. Ibragimov

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 4, pp. 632-655

Résumé

Let $\{\xi_n\}$ be a sequence of independent identically distributed random variables with a common distribution function (d.f.) $F(x)$. Let us assume that d.f. belongs to the domain of attraction of the Gaussian law. Denote by $F_n(x;A_n,B_n)$ the d.f. of normalized sum $S_n=\frac1{B_n}\sum_1^n\xi_i-A_n$ and let $$ \delta_n=\inf_{A_n,B_n}\sup_x|F_n(x;A_n,B_n)-\Phi(x)| $$ where $\Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^xe^{-u^2/2}\,du$. We investigate in this paper the rate of convergence of $\delta_n$ to 0 and some other related problems. The main results which are also indicative of the other results of the paper are the following theorems. Theorem 3.1. {\it In order that $\delta_n=O(n^{-\delta/2})$, $0<\delta<1$, it is necessary and sufficient that the following conditions be satisfied} $$ \sigma^2=\int_{-\infty}^\infty x^2\,dF(x)<\infty,\eqno(3.2) \int_{|x|>z}x^2\,dF(x)=O(|z|^{-\delta}),\quad z\to\infty.\eqno(3.3) $$ Theorem 3.2. {\it In order that $\delta_n=O(n^{-1/2})$ it is necessary and sufficient that conditions (3.1), (3.2) and the following one $$ \int_{-z}^zx^3\,dF(x)=O(1),\quad z\to\infty\eqno(3.4) $$ be satisfied}.