Bernoulli 2 (3), 257-270, (September 1996) DOI: 10.3150/bj/1178291722
Jean-Marc Azaïs, Mario Wschebor
KEYWORDS: crossings of a level, Gaussian processes, Martingales, occupation measure, Stable processes
We show that for various classes of stochastic process, namely Gaussian processes, stable Lévy processes and Brownian martingales, we have almost sure weak convergence of the oscillation in the measure space ([0,1],λ), λ being Lebesgue measure. This result is used to obtain almost sure weak approximation of the occupation measure via numbers of crossings.