On the results of the asymptotic analysis in problems with boundaries

A. A. Borovkov ; V. S. Korolyuk

Teoriâ veroâtnostej i ee primeneniâ, Tome 10 (1965) no. 2, pp. 255-266

Résumé

The paper reviews the results of the asymptotic analysis in the boundary problems for random walks. Let $\xi_1,\xi_2,\dots$ be a sequence of independent identically distributed random variables $S_n=\sum_{k=1}^n\xi_k$ and let $g_n^-(t)$ ($0\le t\le1$) be two functions such that $g_n^\pm(t)/b_n\to g^\pm(t)$ for some $b_n\to\infty$ uniformly on $[0,1]$. Let $\eta_g$ be the first passade time of the random trajectory $\{k/n,S_k\}$, $k=\overline{1,n}$ out of the region $g_n$ contained between the curves $x=g_n^\pm(t)$, $0\le t\le1$: $$ \eta_g=1+\max\biggl\{k\colon g_n^-\biggl(\frac jn\biggr)^+\biggl(\frac jn\biggr),\quad j=0,1,\dots,k\le n\biggr\} $$ and $\chi_g$ be the value of the first jump over the boundary of $g_n$. The content of the article is the review of the results on limit theorems for the joint distributions of random variables $\eta_g$, $\chi_g$, $S_n$ as $n\to\infty$. The distributions of some other functionals of the trajectory $S_k$, $k=\overline{1,n}$ are also considered.