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Summary: Probability on Compact Lie Groups
by David Applebaum, University of Sheffield 1
Prepared for forthcoming INTERNATIONAL ENCYCLOPEDIA OF
STATISTICAL SCIENCES, to be published by Springer
1. Introduction. Probability on groups enables us to study the inter-
action between chance and symmetry. In this article I'll focus on the case
where symmetry is generated by continuous groups, specifically compact Lie
groups. This class contains many examples such as the n-torus, special or-
thogonal groups SO(n) and special unitary groups SU(n) which are impor-
tant in physics and engineering applications. It is also a very good context to
demonstrate the key role played by non-commutative harmonic analysis via
group representations. The classic treatise [4] by Heyer gives a systematic
mathematical introduction to this topic while Diaconis [2] presents a wealth
of concrete examples in both probability and statistics.
For motivation, let be a probability measure on the real line. Its charac-
teristic function is the Fourier transform (u) = R
eiux
(dx) and uniquely
determines . Note that the mappings x eiux
are the irreducible unitary
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