%0 Journal Article
%T Grey Brownian motion local time: Existence and weak-approximation
%A Jos¨¦ Lu¨ªs Da Silva
%A Mohamed Erraoui
%J Mathematics
%D 2013
%I arXiv
%R 10.1080/17442508.2014.945451
%X In this paper we investigate the class of grey Brownian motions $B_{\alpha,\beta}$ ($0<\alpha<2$, $0<\beta\leq1$). We show that grey Brownian motion admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. The weak convergence of the increments of $B_{\alpha,\beta}$ in $t$, $w$-variables are studied. Using the Berman criterium we show that $B_{\alpha,\beta}$ admits a $\lambda$-square integrable local time $L^{B_{\alpha,\beta}}(\cdot,I)$ almost surely ($\lambda$ Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings $C^{B_{\alpha,\beta}^{\varepsilon}}(x,I)$, of level $x$, of the convolution approximation $B_{\alpha,\beta}^{\varepsilon}$ of grey Brownian motion.
%U http://arxiv.org/abs/1306.3956v1