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February, 1977 Level Crossings of a Stochastic Process with Absolutely Continuous Sample Paths
Michael B. Marcus
Ann. Probab. 5(1): 52-71 (February, 1977). DOI: 10.1214/aop/1176995890

Abstract

Let $X(t), t \in \lbrack 0, 1 \rbrack$ be a real valued stochastic process with absolutely continuous sample paths. Let $M(a, X(t))$ denote the number of times $X(t) = a$ for $t \in (0, 1\rbrack$ and $N(a, X(t))$ the number of times $X(t)$ crosses the level $a$ for $t \in (0, 1\rbrack$. Under certain conditions on the joint density function of $X(t)$ and its derivative $X(t)$, integral expressions are obtained for $E \lbrack \prod^k_{i = 1} N(a_i, X(t))^j_i \rbrack$ for $j_i$ positive integers (similarly with $M$ replacing $N$). Examples of Gaussian processes $X(t)$ are found for which $X(0) \equiv 0, EN(a, X(t)) < \infty, a \neq 0$ but $EN(0, X(t)) = \infty$. Also examples of stationary Gaussian processes are given for which $EN(a, X(t)) < \infty$ for all $a, EN^2(0, X(t)) = \infty$ but $E\rbrack N(0, X(t))N(a, X(t)) \rbrack < \infty$ for $a \neq 0$. These examples are used to describe the clustering of the zeros of a certain class of Gaussian processes.

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Michael B. Marcus. "Level Crossings of a Stochastic Process with Absolutely Continuous Sample Paths." Ann. Probab. 5 (1) 52 - 71, February, 1977. https://doi.org/10.1214/aop/1176995890

Information

Published: February, 1977
First available in Project Euclid: 19 April 2007

zbMATH: 0364.60063
MathSciNet: MR426124
Digital Object Identifier: 10.1214/aop/1176995890

Subjects:
Primary: 60G17
Secondary: 60G15 , 60H99

Keywords: absolutely continuous sample paths , clustering of zeros , counting function , Gaussian processes , level crossings

Rights: Copyright © 1977 Institute of Mathematical Statistics

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Vol.5 • No. 1 • February, 1977
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