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