Asymptotics of Randomly Stopped Sequences with Independent Increments

Abstract

Let $S_n, n = 1, 2, \cdots$, be a sequence of sums of independent, identically distributed random variables $X_i$ such that $P\{X_i > y\}$ is a regularly varying function of $y$ at infinity. Let $N$ be a stopping time for $S_n$ with finite mean. A necessary and sufficient condition is given that $$\lim_{y\rightarrow \infty} P\{S_N > y\}/P\{X_1 > y\} = EN.$$ Examples further illustrate the role of this condition.