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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Infinite convolutions on locally compact Abelian groups and additive functions
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by Philip Hartman PDF
Trans. Amer. Math. Soc. 214 (1975), 215-231 Request permission

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.
References
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 214 (1975), 215-231
  • MSC: Primary 60B15; Secondary 10K99, 43A05
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0400333-9
  • MathSciNet review: 0400333