On time- and cycle-stationarity
A process with cycles in a pair \((Z,S)\), where \(Z = (Z_ s)_{s \in \mathbb{R}}\) is a stochastic process and \(S = (S_ k)_{k \in \mathbb{Z}}\) is a sequence of random times satisfying \(-\infty \leftarrow \cdots < S_{-2} < S_{-1} < 0 \leq S_ 0 < S_ 1 < S_ 2 < \cdots \to \infty\). We call \((Z,S)\) time-stationary resp. cycle-stationary iff for all \(t \in \mathbb{R}\) resp. all \(n \in \mathbb{Z}\) the processes \(\theta_ t (Z,S)\) resp. \(\theta_{S_ n} (Z,S)\) and \((Z,S)\) have the same distribution. Here \(\theta_ u (Z,S)\) denotes the process \((Z,S)\) centered at \(u\). By using standard Palm methods the following (informal) relationships between time-stationary and cycle-stationary processes are shown: The cycle- stationary process behaves like the time-stationary one conditioned on a point at 0; the cycle-stationary process behaves like the time-stationary one centered at a cycle point chosen at random; the time-stationary process behaves like the cycle-stationary one centered at a time in \((- \infty, \infty)\) chosen at random.
scientific article; zbMATH DE number 735481
scientific article