Extreme Time of Stochastic Processes with Stationary Independent Increments

Abstract

Let $\{S_n = \sum^n_{i=1} Y_i\}$ or $\{X_t, t \geqq 0\}$ be a stochastic process with stationary independent increments, and let $T^+(\tau), T^-(\tau)$ be the times elapsed until the process has spent time $\tau$ at its maximum and minimum respectively, defined in terms of local time when necessary. Bounds in terms of moments of $Y_1$ or $X_1$ are given for $E(\min (T^+(\tau), T^-(\tau)))$. The discrete case is studied first and the result for continuous-time processes is obtained by a limiting argument. As an auxiliary it is shown that the local time at zero of a process $X_t$ minus its maximum can be approximated uniformly in probability using the number of new maxima attained by the process observed at discrete times.