Spectral representations of characteristic functions of discrete probability laws

Abstract

We consider discrete probability laws on the real line, whose characteristic functions are separated from zero. In particular, this class includes arbitrary discrete infinitely divisible laws and lattice probability laws, whose characteristic functions have no zeroes on the real line. We show that characteristic functions of such laws admit spectral Lévy-Khinchine type representations. We also apply the representations of such laws to obtain limit and compactness theorems with convergence in variation to probability laws from this class. Thus we generalize the corresponding results from the recent papers [11] and [17].

A criterion of quasi-infinite divisibility for discrete laws

“…|f (t)| µ > 0 for some µ and all t ∈ R, then the law is quasi-infinitely divisible (see the next section for more details). The latter result generalizes the previous, because in the lattice case the function t → |f (t)|, t ∈ R, is continuous and periodic, and the absence of zeroes is equivelent to the separatness from zero over the period segment (see [1] for more details). Anyway, however, the following interesting question remains here (it was posed in [1]).…”

“…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).…”

“…This note is devoted to the question posed in [1] about the description of quasi-infinitely divisibile laws within the class of arbitrary univariate discrete probability laws.…”

“…It is rather natural and interesting to investigate criteria of quasi-infinite divisibility for probability laws. The most deep results here were obtained for univariate discrete laws in [1], [9], and [12]. In particular, following [12], a discrete lattice probability law is quasi-infinitely divisible if its characteristic function f has no zeroes on the real line, i.e.…”

“…In particular, following [12], a discrete lattice probability law is quasi-infinitely divisible if its characteristic function f has no zeroes on the real line, i.e. f (t) = 0, t ∈ R. For arbitrary discrete laws only the sufficient condition is known due to [1]. Namely, if f is separated from zero, i.e.…”