Abstract
We investigate the behaviour of the entropy of convolutions of independent random variables on compact groups. We provide an explicit exponential bound on the rate of convergence of entropy to its maximum. Equivalently, this proves convergence of the density to uniformity, in the sense of Kullback–Leibler. We prove that this convergence lies strictly between uniform convergence of densities (as investigated by Shlosman and Major), and weak convergence (the sense of the classical Ito–Kawada theorem). In fact it lies between convergence in L 1+ε and convergence in L 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Barron, A. R. (1986). Entropy and the central limit theorem. Ann. Prob. 14, 336–342.
Csiszár, I. (1964). A note on limiting distributions on topological groups. Matematikai Kutató Intézetenek Közleményei 9, 595–598.
Derriennic, Y. (1985). Entropie, theoreèmes limites et marches aléatoires. In Heyer, H. (ed.), Probability Measures on Groups VIII, Oberwolfach, Lecture Notes in Mathematics, Vol. 1210, Springer-Verlag, Berlin. pp. 241–284.
Grenander, U. (1963). Probabilities on Algebraic Structures, Wiley, New York.
Heyer, H. (1977). Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin.
Kloss, B. M. (1959). Probability distributions on bicompact topological groups. Theory of Probability and Its Applications 4, 237–270.
Kullback, S. (1967). A lower bound for discrimination information in terms of variation. IEEE Transactions on Information Theory 13, 126–127.
Major, P., and Shlosman, S. B. (1979). A local limit theorem for the convolution of probability measures on a compact connected group. Zeitschrift für Wahrscheinlichkeits-theorie und verwandte Gebiete 50, 137–148.
Saloff-Coste, L. (1997). Lectures on finite Markov Chains. In Bernard, P. (ed.), Lectures on Probability Theory and Statistics, St-Flour 1996, Lecture Notes in Mathematics, Vol. 1665, Springer-Verlag, Berlin. pp. 301–413.
Shlosman, S. B. (1980). Limit theorems of probability theory on compact topological groups. Theory of Probability and Its Applications 25, 604–609.
Shlosman, S. B. (1984). The influence of non-commutativity on limit theorems. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 65, 627–636.
Stromberg, K. (1960). Probabilities on a compact group. Trans. Amer. Math. Soc. 94, 295–309.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Johnson, O., Suhov, Y. Entropy and Convergence on Compact Groups. Journal of Theoretical Probability 13, 843–857 (2000). https://doi.org/10.1023/A:1007818830500
Issue Date:
DOI: https://doi.org/10.1023/A:1007818830500