On the rate of approach of the distributions of sums of independent random variables to accompanying distributions

I. A. Ibragimov ; E. L. Presman

Teoriâ veroâtnostej i ee primeneniâ, Tome 18 (1973) no. 4, pp. 753-766

Résumé

Let $\mathcal{F}$ be the set of all distribution functions on $R,\mathcal{F}^*$ the subset of $\mathcal{F}$ corresponding to symmetric random variables, $F^n$ $n$-times convolution of $F$ with itself, $E_a$ the distribution function corresponding to the unit mass at $a$, $|F-G|=\sup_x |F(x)-G(x)|$ for $F,G\in\mathcal{F}$. It is proved that $$ \frac{c_0}{n^{1/3}}\sup_{F\in\mathcal{F}}\inf_a |(E_a F)^n-\exp\{n(E_a F-E_0)\}|\leq\frac{8}{n^{1/3}}, $$ $$ \frac{c_1}{\sqrt{n}}\sup_{F\in\mathcal{F}^*}|F^n-\exp\{n(F-E_0)\}|\sqrt{\frac{\log n}{n}}. $$ Here the first right-hand inequality is Kolmogorov's uniform limit theorem in Le Cam's version. We study also the closeness of distribution functions $\prod_i F_i E_{a_i}$ and $\exp\sum_i (F_iE_{a_i}-E_0)$ in the Kolmogorov–Smirnov and Lévy metrices.