Infinite convolutions on locally compact Abelian groups and additive functions

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.