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Probability Measures on Compact Groups which have Square-Integrable Densities

Summary: Probability Measures on Compact Groups
which have Square-Integrable Densities
David Applebaum
We apply Peter-Weyl theory to obtain necessary and sufficient
conditions for a probability measure on a compact group to have a
square-integrable density. Applications are given to measures on the
d-dimensional torus.
MSC 2000: 60B15, 60E07, 43A05, 43A30
1 Introduction
Given a probability measure on a Riemannian manifold it is highly beneficial
if we can establish that it has a density with respect to the Riemannian
volume measure. This is vital for statistical inference but a density may
also contain important information about the topology and geometry of the
manifold, e.g. the heat kernel is the density of Brownian motion on a manifold
(see e.g. [6], [18].) Much work by stochastic analysts has been focussed
on finding densities for solutions of stochastic differential equations driven
by Brownian motion. A key result here is H¨ormander's theorem [10] which
gives a sufficient condition (called "hypoellipticity") for existence of a smooth
density involving the Lie algebra generated by the driving vector fields. The


Source: Applebaum, David - Department of Probability and Statistics, University of Sheffield


Collections: Mathematics