A Note on Occupation Times of Stationary Processes
Marina Kozlova, Abo Akademi University, Finland
Paavo Salminen, Abo Akademi University, Finland
Abstract
Consider a real valued stationary process $X={X_s:, sinRR}$.
For a fixed $tin RR$ 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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