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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Infinite convolutions on locally compact Abelian groups and additive functions

Author: Philip Hartman
Journal: Trans. Amer. Math. Soc. 214 (1975), 215-231
MSC: Primary 60B15; Secondary 10K99, 43A05
MathSciNet review: 0400333
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Abstract: Let ${\mu _1},{\mu _2}, \ldots$ be regular probability measures on a locally compact Abelian group G such that $\mu = {\mu _1} \ast {\mu _2} \ast \cdots = \lim {\mu _1} \ast \cdots \ast {\mu _n}$ exists (and is a probability measure). For arbitrary G, we derive analogues of the Lévy theorem on the existence of an atom for $\mu$ and of the “pure theorems” of Jessen, Wintner and van Kampen (dealing with discrete ${\mu _1},{\mu _2}, \ldots$) in the case $G = {R^d}$. These results are applied to the asymptotic distribution $\mu$ of an additive function $f:{Z_ + } \to G$ after generalizing the Erdös-Wintner result $(G = {R^1})$ which implies that $\mu$ is an infinite convolution of discrete probability measures.

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Keywords: Locally compact Abelian groups, regular probability measures, infinite convolutions, absolutely continuous, purely discontinuous (= discrete), singular measures, additive functions, asymptotic distributions
Article copyright: © Copyright 1975 American Mathematical Society