On a denseness result for quasi-infinitely divisible distributions

Abstract

A probability distribution µ on R d is quasi-infinitely divisible if its characteristic function has the representation µ = µ 1 / µ 2 with infinitely divisible distributions µ 1 and µ 2 . In [6, Thm. 4.1] it was shown that the class of quasi-infinitely divisible distributions on R is dense in the class of distributions on R with respect to weak convergence. In this paper, we show that the class of quasi-infinitely divisible distributions on R d is not dense in the class of distributions on R d with respect to weak convergence if d ≥ 2.

A criterion of quasi-infinite divisibility for discrete laws

“…The first detailed analysis of quasi-infinitely divisible laws based on their Lévy type representations was performed in [12], where the authors studied questions concerning supports, moments, continuity, and weak convergence for these laws. These results were generalized and complemented in the papers [1], [2], [3], [4], [5], [8], [9] and [10]. The quasi-infinitely divisible laws have already found a lot of interesting applications (see [5], [6], [13], [16], [17] and the references given there).…”