309 research outputs found
Continuous Association Schemes and Hypergroups
Classical finite association schemes lead to a finite-dimensional algebras
which are generated by finitely many stochastic matrices. Moreover, there exist
associated finite hypergroups. The notion of classical discrete association
schemes can be easily extended to the possibly infinite case. Moreover, the
notion of association schemes can be relaxed slightly by using suitably
deformed families of stochastic matrices by skipping the integrality
conditions. This leads to larger class of examples which are again associated
to discrete hypergroups.
In this paper we propose a topological generalization of the notion of
association schemes by using a locally compact basis space and a family of
Markov-kernels on indexed by a further locally compact space where the
supports of the associated probability measures satisfy some partition
property. These objects, called continuous association schemes, will be related
to hypergroup structures on . We study some basic results for this new
notion and present several classes of examples. It turns out that for a given
commutative hypergroup the existence of an associated continuous association
scheme implies that the hypergroup has many features of a double coset
hypergroup. We in particular show that commutative hypergroups, which are
associated with commutative continuous association schemes, carry dual positive
product formulas for the characters. On the other hand, we prove some rigidity
results in particular in the compact case which say that for given spaces
there are only a few continuous association schemes
Central limit theorems for multivariate Bessel processes in the freezing regime
Multivariate Bessel processes are classified via associated root systems and
positive multiplicity constants. They describe the dynamics of interacting
particle systems of Calogero-Moser-Sutherland type. Recently, Andraus, Katori,
and Miyashita derived some weak laws of large numbers for these processes for
fixed positive times and multiplicities tending to infinity. In this paper we
derive associated central limit theorems for the root systems of types A, B and
D in an elementary way. In most cases, the limits will be normal distributions,
but in the B-case there are freezing limits where distributions associated with
the root system A or one-sided normal distributions on half-spaces appear. Our
results are connected to central limit theorems of Dumitriu and Edelman for
beta-Hermite and beta-Laguerre ensembles
Product formulas for a two-parameter family of Heckman-Opdam hypergeometric functions of type BC
In this paper we present explicit product formulas for a continuous
two-parameter family of Heckman-Opdam hypergeometric functions of type BC on
Weyl chambers of type . These formulas are related
to continuous one-parameter families of probability-preserving convolution
structures on . These convolutions on
are constructed via product formulas for the spherical functions of the
symmetric spaces and associated double coset
convolutions on with the torus . We shall
obtain positive product formulas for a restricted parameter set only, while the
associated convolutions are always norm-decreasing. Our paper is related to
recent positive product formulas of R\"osler for three series of Heckman-Opdam
hypergeometric functions of type BC as well as to classical product formulas
for Jacobi functions of Koornwinder and Trimeche for rank
Dispersion and limit theorems for random walks associated with hypergeometric functions of type BC
The spherical functions of the noncompact Grassmann manifolds
over the (skew-)fields with rank and dimension parameter can be described
as Heckman-Opdam hypergeometric functions of type BC, where the double coset
space is identified with the Weyl chamber of
type B. The corresponding product formulas and Harish-Chandra integral
representations were recently written down by M. R\"osler and the author in an
explicit way such that both formulas can be extended analytically to all real
parameters , and that associated commutative convolution
structures on exist. In this paper we introduce moment functions
and the dispersion of probability measures on depending on and
study these functions with the aid of this generalized integral representation.
Moreover, we derive strong laws of large numbers and central limit theorems for
associated time-homogeneous random walks on where the moment
functions and the dispersion appear in order to determine drift vectors and
covariance matrices of these limit laws explicitely. For integers , all
results have interpretations for -invariant random walks on the
Grassmannians .
Besides the BC-cases we also study the spaces , which are related to Weyl chambers of type A, and for which corresponding
results hold. For the rank-one-case , the results of this paper are
well-known in the context of Jacobi-type hypergroups on .Comment: Extended version of arXiv:1205.4866; some corrections to prior
version. Accepted for publication in J. Theor. Proba
Olshanski spherical functions for infinite dimensional motion groups of fixed rank
Consider the Gelfand pairs associated
with motion groups over the fields
with and fixed as well as the inductive limit ,the
Olshanski spherical pair . We classify all Olshanski
spherical functions of as functions on the cone
of positive semidefinite -matrices and show that they appear as
(locally) uniform limits of spherical functions of as .
The latter are given by Bessel functions on . Moreover, we determine all
positive definite Olshanski spherical functions and discuss related positive
integral representations for matrix Bessel functions. We also extend the
results to the pairs which
are related to the Cartan motion groups of non-compact Grassmannians. Here
Dunkl-Bessel functions of type B (for finite ) and of type A (for
) appear as spherical functions
A Limit Relation for Dunkl-Bessel Functions of Type A and B
We prove a limit relation for the Dunkl-Bessel function of type with
multiplicity parameters on the roots and on where tends to infinity and the arguments are suitably scaled. It
gives a good approximation in terms of the Dunkl-type Bessel function of type
with multiplicity . For certain values of an improved
estimate is obtained from a corresponding limit relation for Bessel functions
on matrix cones.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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