J. Aust. Math. Soc.
73 (2002), 301-333
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Limit theorems for isotropic random walks on triangle buildings
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Marc Lindlbauer
Mathematisches Institut
Universität Tübingen
Auf der Morgenstelle 10
72076 Tübingen
Germany
and
GSF-Forschungszentrum
für Umwelt und Gesundheit
85764 Neuherberg
Germany
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Abstract
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The spherical functions of triangle buildings
can be described in terms of certain
two-dimensional orthogonal polynomials on
Steiner's hypocycloid which are closely related
to Hall-Littlewood polynomials. They lead to a
one-parameter family of two-dimensional
polynomial hypergroups. In this paper we
investigate isotropic random walks on the vertex
sets of triangle buildings in terms of their
projections to these hypergroups. We present
strong laws of large numbers, a central limit
theorem, and a local limit theorem; all these
results are well-known for homogeneous trees.
Proofs are based on moment functions on
hypergroups and on explicit expansions of the
hypergroup characters in terms of certain
two-dimensional Tchebychev polynomials.
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