The Annals of Probability

Level Crossings of a Stochastic Process with Absolutely Continuous Sample Paths

Michael B. Marcus

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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.

Article information

Ann. Probab., Volume 5, Number 1 (1977), 52-71.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G17: Sample path properties
Secondary: 60H99: None of the above, but in this section 60G15: Gaussian processes

Level crossings absolutely continuous sample paths clustering of zeros Gaussian processes counting function


Marcus, Michael B. Level Crossings of a Stochastic Process with Absolutely Continuous Sample Paths. Ann. Probab. 5 (1977), no. 1, 52--71. doi:10.1214/aop/1176995890.

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