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Finite Gel’fand pairs and their applications to probability and statistics. (English. Russian original) Zbl 1173.43001
J. Math. Sci., New York 141, No. 2, 1182-1229 (2007); translation from Sovrem. Mat. Prilozh. 27, 95-140 (2005).
This paper gives a survey about finite Gelfand pairs, spherical functions, and applications to Markov chains. It may be regarded as a preliminary version or, in some respect, counterpart of the recent book of the authors [Harmonic analysis on finite groups. Representation theory, Gelfand pairs and Markov chains. Cambridge: Cambridge University Press (2008; Zbl 1149.43001)].
The first part of the paper contains a general introduction to finite Gelfand pairs and the associated spherical functions, and is very similar to classical introductions to general Gelfand pairs. In the second part, the authors review details about three classical important classes of examples, namely the Hamming schemes and Krawtchouk polynomials, the Johnson schemes and Hahn polynomials, and finally the higher rank examples $$(S_{2n}, S_2\wr S_n)$$. In the final part, the authors show how these machine can be applied to the rate of convergence of associated random walks to their stationary distributions in the spirit of P. Diaconis. Here in particular, the Ehrenfest urn model, the Laplace-Bernoulli urn model as well as some party model are considered.

##### MSC:
 43A05 Measures on groups and semigroups, etc. 43A90 Harmonic analysis and spherical functions 33C80 Connections of hypergeometric functions with groups and algebras, and related topics 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60B10 Convergence of probability measures
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