On the exit of random walk out of the curvilinear domain

M. U. Gafurov ; V. I. Rotar'

Teoriâ veroâtnostej i ee primeneniâ, Tome 28 (1983) no. 1, pp. 169-175

Résumé

Let $S_0=0$, $S_n=X_1+\dots+X_n$, where $X_1,X_2,\dots$ are i. i. d. r. v.'s; let $\varphi(x)$, $g(x)$ be regularly varying strictly increasing positive functions and $g(n)\ n^{-1/2}\to\infty$, $n\to\infty$. Let $N_g=\min\{n\colon S_n>g(n)\}$, $S_g=S_{N_g}$, $\chi_g=S_g-g(N_g)$, $q_\infty=\mathbf P\{|S_k|\le g(k)\ \forall\,k\}$. The typical result of the paper is the following Theorem. {\it Let for any $c>0$ $$ \sum_{n=1}^\infty\varphi(g(n))n^{-1}\exp\{-cg^2(n)\,n^{-1}\}<\infty. $$ Then $\mathbf E[\varphi(S_g)\mid N_g<\infty]<\infty$ if (and in the case $q_\infty>0$ only if) $$ \mathbf E\varphi(X_1^+)G(X_1^+)<\infty,\qquad\text{where}\quad G=g^{-1}. $$} The analogous results are obtained for $N_g$, $\chi_g$.