Central limit theorem and infinitely divisible probabilities associated with partial differential operators

Abstract

We provide $$K_\ell = \left\{ {\begin{array}{*{20}c} {[0,\infty [ \times [ - \pi \ell ,\pi \ell ] if \ell \in \mathbb{N}\backslash \{ 0\} } \\ {[0, + \infty [ \times \mathbb{R} if \ell = 0} \\ \end{array} } \right.$$ with a generalized convolution product * associated with a system of partial differential operators. We prove that (Kl, *) is an hypergroup in the sense of Jewett only in the case l=1. Next we establish a central limit theorem onKl for all l∈ℕ and we give a Levy-Kintchine formula which characterizes the convolution semigroups and the infinitely divisible probabilities onKl.