On Convergence and Compactness in Variation with a Shift of Discrete Probability Laws

Abstract

We consider a class of discrete distribution functions whose characteristic functions are separated from zero, i.e., their absolute values are larger than a certain positive constant on the entire real line. This class is rather wide: it contains discrete infinitely divisible distribution functions, functions of lattice distributions whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently, we demonstrated in our study that characteristic functions corresponding to elements of this class admit the Levy–Khinchin type representation with a nonmonotonic spectral function. Thus, our class is included in the set of quasi-infinitely divisible distribution functions. Using these representations, we also obtained limit theorems and theorems of compactness with convergence in variation for the sequences from this class. This paper is devoted to analogous results concerning convergence and compactness, but with somewhat weakened convergence in variation. Changing the type of convergence notably expands the applicability of the results.