The accuracy of approximation oi the limit distribution to the distribution of the maximum of sums of independent random variables

B. A. Rogozin

Teoriâ veroâtnostej i ee primeneniâ, Tome 11 (1966) no. 3, pp. 497-500

Résumé

Let $\xi_1\xi_2,\dots$ be a sequence of identically distributed independent random variables n and $S_0=0$, $S_n=\sum_{k=1}^n\xi_k$, $n=1,2,\dots$, $\bar S_n=\max_{0\le k\le n}S_k$, $n=0,1\dots$. Let us suppose that $\mathbf M\xi_1=a>0$, $\beta_3=\mathbf M|\xi_1-a|^3<\infty$, and denote $\sigma^2=\mathbf M(\xi_1-a)^2$. It is established that $$ \mathbf P\{S_n\le x\}-\mathbf P\{\bar S_n\le x\}\le\frac C{\sqrt n}\max\biggl\{\frac{\beta_3^2}{\sigma^6},\frac{\beta_3^2}{a^6},\frac{(\mathbf M|\xi_1|)^2}{\sigma^2}\biggr\} $$ where $С$ is a constant.