Level Crossings of a Stochastic Process with Absolutely Continuous Sample Paths

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.