Abstract
Consider a sphere of radius rootn 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.
Translated title of the contribution | Entropy and a generalization of Poincare's observation |
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Original language | English |
Pages (from-to) | 375 - 384 |
Number of pages | 10 |
Journal | Mathematical Proceedings of the Cambridge Philosophical Society |
Volume | 135 (2) |
DOIs | |
Publication status | Published - Sep 2003 |
Bibliographical note
Publisher: Cambridge Univ PressOther identifier: ISD number 725QC