Mixtures in nonstable Lévy processes

Abstract

We analyse the Lévy processes produced by means of two interconnected classes of nonstable, infinitely divisible distribution: the variance gamma and the Student laws. While the variance gamma family is closed under convolution, the Student one is not: this makes its time evolution more complicated. We prove that—at least for one particular type of Student processes suggested by recent empirical results, and for integral times—the distribution of the process is a mixture of other types of Student distributions, randomized by means of a new probability distribution. The mixture is such that along the time the asymptotic behaviour of the probability density functions always coincide with that of the generating Student law. We put forward the conjecture that this can be a general feature of the Student processes. We finally analyse the Ornstein-Uhlenbeck process driven by our Lévy noises and show a few simulations of it.