Random Stopping Preserves Regular Variation of Process Distributions

Let $S_n$ be a stochastic process with either discrete or continuous time parameter and stationary independent increments. Let $N$ be a stopping time for the process such that $EN < \infty$. If the upper tail of the process distribution, $F$, is regularly varying, certain conditions on the lower tail of $F$ and on the tail of the distribution of $N$ imply that $\lim_{y\rightarrow\infty}P(S_N > y)/(1 - F(y)) = EN$. A similar asymptotic relation is obtained for $\sup_n S_{n \wedge N}$, if $n$ is discrete. These asymptotic results are related to the Wald moment identities and to moment inequalities of Burkholder. Applications are given for exit times at fixed and square-root boundaries.