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August, 1975 Extreme Time of Stochastic Processes with Stationary Independent Increments
Priscilla Greenwood
Ann. Probab. 3(4): 664-676 (August, 1975). DOI: 10.1214/aop/1176996307

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.

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Priscilla Greenwood. "Extreme Time of Stochastic Processes with Stationary Independent Increments." Ann. Probab. 3 (4) 664 - 676, August, 1975. https://doi.org/10.1214/aop/1176996307

Information

Published: August, 1975
First available in Project Euclid: 19 April 2007

zbMATH: 0333.60055
MathSciNet: MR402937
Digital Object Identifier: 10.1214/aop/1176996307

Subjects:
Primary: 60G40
Secondary: 60J55

Keywords: Local time , Maximal process , moment conditions , Random walk , stopping times

Rights: Copyright © 1975 Institute of Mathematical Statistics

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Vol.3 • No. 4 • August, 1975
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