Entropy and a generalisation of `Poincare's Observation'

Details Entropy and a generalisation of `Poincare's Observation' Entropy and a generalisation of `Poincare's Observation'

2002-01-29

arxiv.org/abs/math/0201273v1

Mathematical Proceedings of the Cambridge Philosophical Society, Vol 135/2, 2003, pages 375-384

Mathematics

Probability

12 pages

Scientific paper

10.1017/S0305004103006881

Consider a sphere of radius root(n) in n dimensions, and consider X, a random variable uniformly distributed on its surface. Poincare's Observation states that for large n, the distribution of the first k coordinates of X is close in total variation distance to the standard normal N(0,I_k). In this paper, we consider a larger family of manifolds, and X taking a more general distribution on the surfaces. We establish a bound in the stronger Kullback--Leibler sense of relative entropy, and discuss its sharpness, providing a necessary condition for convergence in this sense. We show how our results imply the equivalence of ensembles for a wider class of test functions than is standard. We also deduce results of de Finetti type, concerning a generalisation of the idea of orthogonal invariance.

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