An estimation of a~convergence rate for the absorption probability in case of a~null expectation

S. V. Nagaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 13 (1968) no. 1, pp. 160-164

Résumé

Let $\xi_1,\xi_2,\dots$ be a sequence of mutually independent equally distributed random variables with a distribution function $F_\lambda(x)$ depending on a parameter $\lambda$. Let $\mathbf M\xi_1^2=2\lambda^2$ and $\mathbf M\xi_1=0$. Define $n_x$ as the least integer for which $\zeta_n+x\notin(a,b)$, where $\zeta_n=\sum_{i=1}^n\xi_i$ and $(a,b)$ is a finite interval of the real line. Put $$ P_\lambda(x)=\mathbf P\{\zeta_{n_x}+x\ge b\},\quad x\in(a,b), $$ and $$ c_{3\lambda}=\mathbf M|\xi_1|^3. $$ The following assertion is proved: there exists an absolute constant $L$ such that $$ \sup_{a}\biggl|P_\lambda(x)-\frac{x-a}{b-a}\biggr|\frac{c_{3\lambda}}{\lambda^2(b-a)}. $$