Generalized grey Brownian motion local time: existence and weak approximation

Abstract

In this paper we investigate the class of generalized grey Brownian motions (ggBms) (, ). We show that ggBm admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. We establish almost-sure weak convergence of the increments of in the measure space . We also obtain weak convergence of the weighted power variation of process . Using the Berman criterion we show that admits a -square integrable local time almost surely ( denoting Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings , of level x, of the convolution approximation of ggBm.