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A note on occupation times of stationary processes. (English) Zbl 1110.60029

Summary: Consider a real valued stationary process \(X=\{X_s:s\in\mathbb{R}\}\). For a fixed \(t\in\mathbb{R}\) 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. 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.

MSC:

60G10 Stationary stochastic processes
60J55 Local time and additive functionals
60J60 Diffusion processes
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