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Coin tossing and powers of singular measures

Published online by Cambridge University Press:  24 October 2008

Gavin Brown  and
William Moran
Gavin Brown
Affiliation:
University of Liverpool
William Moran
Affiliation:
University of Liverpool
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Extract

We shall be concerned with the probability distributions which arise from naive coin tossing experiments and their repetitions. In each case our main question is whether the resulting measure is singular with respect to Lebesgue measure or absolutely continuous. It is a remarkable fact that althoughthese distributions were defined and studied some forty years ago (see (17)for an interesting account), they already provide examples necessary to recent studies of the convolution measure algebra of a locally compact abeliangroup. In order to verify this one is obliged to make substantial improvements on the classical results but, at least at the technical level, this requires no modern apparatus. Accordingly, we shall restrict attention to the circle. Before doing this, however, we summarize the motivational background from abstract harmonic analysis and indicate the applications of our results to that area.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 77 , Issue 2 , March 1975 , pp. 349 - 364
Copyright
Copyright © Cambridge Philosophical Society 1975

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References

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