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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Limits of successive convolutions


Author: John C. Martin
Journal: Proc. Amer. Math. Soc. 59 (1976), 52-54
MSC: Primary 43A10
DOI: https://doi.org/10.1090/S0002-9939-1976-0412734-X
MathSciNet review: 0412734
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Abstract: On an arbitrary compact, zero-dimensional, Abelian group, if $ {\mu _0},{\mu _1}, \ldots $ is a sequence of probability measures, a condition on these measures is given which is necessary and sufficient for each of the sequences $ {\mu _t},{\mu _t}{\ast}{\mu _{t + 1}},{\mu _t}{\ast}{\mu _{t + 1}}{\ast}{\mu _{t + 2}}, \ldots $ of successive convolutions to converge to Haar measure in the weak-star topology. Some simple consequences of the theorem are noted.


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DOI: https://doi.org/10.1090/S0002-9939-1976-0412734-X
Keywords: Compact zero-dimensional Abelian group, probability measure, convolution, weak-star topology, Fourier-Stieltjes transform
Article copyright: © Copyright 1976 American Mathematical Society