A Note on Occupation Times of Stationary Processes

Marina Kozlova (Abo Akademi University, Finland)
Paavo Salminen (Abo Akademi University, Finland)


Consider a real valued stationary process $X={X_s:, s\in R}$. For a fixed $t\in R$ and a set $D$ in the state space of $X$, let $g_t$ and $d_t$ denote the starting and the ending time, respectively, of an excursion from and to $D$ (straddling $t$). Introduce also the occupation times $I^+_t$ and $I^-_t$ above and below, respectively, the observed level at time $t$ during such an excursion. In this note we show that the pairs $(I^+_t, I^-_t)$ and $(t-g_t, d_t-t)$ are identically distributed. This somewhat curious property is, in fact, seen to be a fairly simple consequence of the known general uniform sojourn law which implies that conditionally on $I^+_t + I^-_t = v$ the variable $I^+_t$ (and also $I^-_t$) is uniformly distributed on $(0,v)$. We also particularize to the stationary diffusion case and show, e.g., that the distribution of $I^-_t+I^+_t$ is a mixture of gamma distributions.

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Pages: 94-104

Publication Date: June 9, 2005

DOI: 10.1214/ECP.v10-1138


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