Almost sure oscillation of certain random processes
For a real-valued random process \(\{X_t: t\in R\}\) and an appropriate normalizing function \(a(\varepsilon)\), \(\varepsilon\geq 0\), define the process \(Z_\varepsilon (t) = (X_{t+ \varepsilon} -X_t)/a (\varepsilon)\), \(t\geq 0\), and the random measure \(\mu(\varepsilon) = {1\over \lambda(I)} \lambda (\{t\in I: Z_\varepsilon(t) \in B\})\), \(B\) Borel set in \(R\), where \(\lambda\) is the Lebesgue measure. The authors show that for a large family of processes the random measures \(\mu_\varepsilon\) converge weakly as \(\varepsilon\to 0\), for almost all elementary events (almost sure) to a nondegenerate measure \(\mu^*\). The results of this kind are proven for the following classes: (1) a class of Gaussian processes including fractional Brownian motion, stationary processes with certain local behaviour, (2) processes with independent increments and symmetric stable law, (3) continuous martingales satisfying some regularity conditions.
scientific article; zbMATH DE number 947103