An estimate of the convergence rate for the absorption probability

S. V. Nagaev

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 1, pp. 140-148

Résumé

Let $\xi_1$, $\xi_2$, …be a sequence of mutually independent equally distributed random variables. Let $\mathbf M\xi_1=m$, $\mathbf M\xi_1^2=2\lambda^2$, $\mathbf M|\xi_1|^3=c_3$. Define $n_x$ as the least integer $n$ for which $\zeta_n+x\notin(a,b)$ where $\zeta_n=\sum_{i=0}^n\xi_i$ and $(a,b)$ is a finite interval of the real line. Put $$ P(x)=P\{\zeta_{n_x}+x\ge b\},\quad x\in(a,b). $$ The following assertion is proved: there exists an absolute constant $L$ such that $$ \sup_{a}|P(x)-u(x)|\frac{Lc_3}{(b-a)\lambda^2}\biggl(1+\frac{|m|}{\lambda^2}(b-a)\biggr) $$ where $u(x)$ is the solution of the equation $$ u''+\frac m{\lambda^2}u'=0 $$ satisfying the boundary conditions $u(a)=0$, $u(b)=1$.