Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T04:39:11.435Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Blum, J. R. & Epstein, B.On the Fourier transforms of an interesting class of measures. Israel J. Math. 10 (1971), 302–305.CrossRefGoogle Scholar
(2)Brown, G.M 0(G) has a symmetric maximal ideal off the Šilov boundary. Proc. London Math. Soc. (3) 27 (1973), 484–504.CrossRefGoogle Scholar
(3)Brown, G.Riesz products and generalised characters. Proc. London Math. Soc. (To appear.)Google Scholar
(4)Brown, G. and Moran, W.A dichotomy for infinite convolutions of discrete measures. Proc. Cambridge Philos. Soc. 73 (1973), 307–316.CrossRefGoogle Scholar
(5)Brown, G. and Moran, W.On orthogonality of Riesz products. Proc. Cambridge Philos. Soc. 76 (1974), 173–181.CrossRefGoogle Scholar
(6)Brown, G. and Moran, W.Sums of random variables in groups and the purity law. Z. Wahrscheinlichkeitstheorie. (To appear.)Google Scholar
(7)Brown, G. and Moran, W.Products of random variables and Kakutani's criterion for orthogonality of product measures. J. London Math. Soc. (To appear.)Google Scholar
(8)Connolly, D. M. and Williamson, J. H. An application of a theory of Singer. (To appear.)Google Scholar
(9)Hewitt, E. and Zuckerman, H. S.Singular measures with absolutely continuous convolution squares. Proc. Cambridge Philos. Soc. 62 (1966), 399–420. Corrigendum Proc. Cambridge Philos. Soc. 63 (1967), 367–8.CrossRefGoogle Scholar
(10)Jessen, B. and Wintner, A.Distribution functions and the Riemann zetafunction. Trans. Amer. Math. Soc. 38 (1935), 48–88.CrossRefGoogle Scholar
(11)Kakutani, S.On equivalence of infinite product measures. Ann. of Math. 49 (1948), 214–224.CrossRefGoogle Scholar
(12)Marsaglia, G.Random variables with independent binary digits. Ann.Math. Statist. 42 (1971),1922–1929.CrossRefGoogle Scholar
(13)Moran, W.The Šilov boundary of M 0(G). Trans. Amer. Math. Soc. 179 (1973), 455–464.Google Scholar
(14)Salem, R.On some singular monotonic functions which are strictly increasing. Trans. Amer. Math. Soc. 53 (1943), 427–439.CrossRefGoogle Scholar
(15)Varopoulos, N. Th.Sur les mesures de Radon d'un groupe localementcompact abelien. C. R. A. Sci. Paris. A 258 (1964), 3805–8, 4896–99; (1965) 1059–62.Google Scholar
(16)Wiener, N. and Wintner, A.On singular distributions. J. Mathematical Phys. 17 (1938), 233–246.CrossRefGoogle Scholar
(17)Wintner, A.The Fourier transforms of probability distributions (Baltimore, The Johns Hopkins University, 1947).Google Scholar