ON NORMALIZERS AND CENTRALIZERS OF COMPACT LIE GROUPS.

APPLICATIONS TO STRUCTURAL PROBABILITY THEORY

Abstract: The concept of operator stability on finite-dimensional vector spaces was
generalized in the past into several directions. In particular, operator-semistable and
self-decomposable laws and self-similar processes were investigated and the underlying
vector space may be replaced by a simply connected nilpotent Lie group This
motivates investigations of certain linear subgroups of GL() and Aut(), respectively,
the decomposability group of a full probability and its compact normal subgroup, the
invariance group.

Using some basic properties of algebraic groups, the structure of normalizers and
centralizers of compact matrix groups is analyzed and applied to the above-mentioned set-up,
proving the existence and describing the shape of exponents and of commuting exponents of
(operator-) semistable laws.

Further applications are mentioned, in particular for operator self-decomposable laws and
self-similar processes.

2000 AMS Mathematics Subject Classification: 22E15, 60B15, 60E07, 60G18.

Key words and phrases: Exponent, commuting exponent, operator-semistable law,
operator-stable law, normalizer, centralizer.