## The Annals of Probability

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

### 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.

#### 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