A Class of Conditional Limit Theorems Related to Ruin Problem

Abstract

For a random walk with mean zero and variance one, a conditional limit theorem is proved under conditions on the path until it for the first time becomes negative. This gives a generalization of a limit theorem for random walk conditioned to stay positive which was considered by Iglehart and others. It has an application to get a tail formula of the d.f. of the maximum for a stopped random walk.