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This article is cited in 2 scientific papers (total in 2 papers)
MATHEMATICS
On convergence and compactness in variation with shift of discrete probability laws
I. A. Alekseeva, A. A. Khartovb a St. Petersburg State University, 7-9, Universitetskaya nab., St. Petersburg, 199034, Russian Federation
b Smolensk State University, 4, ul. Przhevalskogo, Smolensk, 214000, Russian Federation
Abstract:
We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because 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 the authors showed that characteristic functions of elements of this class admit the Lévy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.
Keywords:
characteristic functions, Lévy — Khinchine type representations, quasi-infinitely divisible distributions, convergence in variation, relative compactness, stochastic compactness.
Received: 25.02.2020 Revised: 18.03.2020 Accepted: 19.03.2020
Citation:
I. A. Alekseev, A. A. Khartov, “On convergence and compactness in variation with shift of discrete probability laws”, Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 8:3 (2021), 385–393; Vestn. St. Petersbg. Univ., Math., 8:4 (2021), 221–226
Linking options:
https://www.mathnet.ru/eng/vspua89 https://www.mathnet.ru/eng/vspua/v8/i3/p385
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