Almost sure oscillation of certain random processes. (English) Zbl 0885.60018

Azaïs, Jean-Marc; Wschebor, Mario

Bernoulli 2, No. 3, 257-270 (1996).

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.

Reviewer: W.Szczotka (Wrocław)

MSC:

60F05	Central limit and other weak theorems
60G17	Sample path properties
60G99	Stochastic processes

Keywords:

almost sure oscillation; Gaussian process; fractional Brownian motion; martingales; occupation measure; stable processes; symmetric stable law