The Annals of Probability

Extreme Time of Stochastic Processes with Stationary Independent Increments

Priscilla Greenwood

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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.

Article information

Source
Ann. Probab. Volume 3, Number 4 (1975), 664-676.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176996307

Digital Object Identifier
doi:10.1214/aop/1176996307

Mathematical Reviews number (MathSciNet)
MR402937

Zentralblatt MATH identifier
0333.60055

JSTOR
links.jstor.org

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 60J55: Local time and additive functionals

Keywords
Maximal process local time stopping times random walk moment conditions

Citation

Greenwood, Priscilla. Extreme Time of Stochastic Processes with Stationary Independent Increments. Ann. Probab. 3 (1975), no. 4, 664--676. doi:10.1214/aop/1176996307. https://projecteuclid.org/euclid.aop/1176996307


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