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.