This is a survey lecture on recent developments on random walks with spherical
symmetry: a topic which was opened to research by J.F.C. Kingman in 1963 and
has grown into wide-range applications through the work of W. Hazod, M. Rösler
and M. Voit.
The analytic method to be described in the talk concerns generalized
convolutions of measures on orbit hypergroups, in particular on the self dual
commutative hypergroup of positive semidefinite (hermitian) matrices.
These hypergroups are defined via Bessel functions of higher rank.
From the interplay of various sections of mathematics it becomes evident that
by the very layout of the presentation probabilists, analysts as well as
algebraists may take interest in the subject.
References:
Bloom, Walter R.; Heyer, Herbert:
Harmonic analysis of probability measures on hypergroups.
de Gruyter Studies in Mathematics, 20.
Walter de Gruyter & Co., Berlin, 1995.
Faraut, Jacques; Korányi, Adam:
Analysis on symmetric cones.
Oxford Mathematical Monographs.
Oxford Science Publications.
The Clarendon Press, Oxford University Press, New York, 1994.
Hazod, W.:
Probability on matrix-cone hypergroups: limit theorems and structural
properties.
J. Appl. Anal. 15 (2009), no. 2, 205-245.
Kingman, J. F. C.:
Random walks with spherical symmetry.
Acta Math. 109 (1963) 11-53.
Rösler, Margit:
Bessel convolutions on matrix cones.
Compos. Math. 143 (2007), no. 3, 749-779.
Rösler, Margit; Voit, Michael:
Limit theorems for radial random walks on p x q matrices as p tends to
infinity.
To appear in Math. Nachr.
Voit, Michael:
Bessel convolutions on matrix cones: algebraic properties and random walks.
J. Theoret. Probab. 22 (2009), no. 3, 741-771.
Voit, Michael:
Central limit theorems for radial random walks on p x q matrices for p tends
to infinity.
To appear in Proc. Amer. Math. Soc.
Zeuner, Hansmartin:
Moment functions and laws of large numbers on hypergroups.
Math. Z. 211 (1992), no. 3, 369-407.