Time to reach stationarity in the Bernoulli-Laplace diffusion model. (English) Zbl 0617.60009

Diaconis, Persi; Shahshahani, Mehrdad

SIAM J. Math. Anal. 18, 208-218 (1987).

For the Bernoulli-Laplace diffusion model involving n balls in each urn, with stationary distribution \(\pi_ n\), the authors study the variation distance \[ \| P_ k-\pi_ n\| =(1/2)\sum_{j}| P_ k(j)- \pi_ n(j)| \] of the law of the process after k steps. It turns out that \(\| P_ k-\pi_ n\| \leq ae^{-2c}\) with \(k:=(1/4)n\log n+cn\), \(c\geq 0\) being a universal constant, and that \(\| P_ k-\pi_ n\| \geq 1-be^{4c}\) with k as above, \(c\in [(-1/4)\log n\), 0], \(b>0\) universal.
In fact, the authors deal with the slightly more general case that the one urn contains r red balls and the other one n-r black balls. The method of treating this problem is an application of the theory of spherical functions on the homogeneous space \(S_ n/S_ r\times S_{n- r}\) [dual Hahn polynomials; S. Karlin and J. McGregor, The Hahn polynomials, formulas and an application. Scripta Math. 26, 33-46 (1961; Zbl 0104.291)], where \(S_ n\) denotes the symmetric group of order n. It is also shown that this method can be applied to nearest neighbor random walks on two-point homogeneous spaces (whose spherical functions are orthogonal polynomials [D. Stanton, Special functions: Group theoretical aspects and applications, Math. Appl., D. Reidel Publ. Co. 18, 87-128 (1984; Zbl 0578.20041)]), in particular on the m-dimensional cube and on the space of k-dimensional subspaces of a vector space over a finite field.

Reviewer: H.Heyer

MSC:

60B15	Probability measures on groups or semigroups, Fourier transforms, factorization
60G50	Sums of independent random variables; random walks
43A90	Harmonic analysis and spherical functions

Keywords:

Gelfand pairs; Bernoulli-Laplace diffusion; spherical functions; nearest neighbor random walks