Abstract
We bound the rate of convergence to stationarity for a signed generalization of the Bernoulli–Laplace diffusion model; this signed generalization is a Markov chain on the homogeneous space (\({\mathbb{Z}}\) 2≀S n )/(S r ×S n−r ). Specifically, for r not too far from n/2, we determine that, to first order in n, \(\tfrac{1}{4}\) n log n steps are both necessary and sufficient for total variation distance to become small. Moreover, for r not too far from n/2, we show that our signed generalization also exhibits the “cutoff phenomenon.”
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REFERENCES
Diaconis, P. (1988). Group Representations in Probability and Statistics, Institute of Mathematical Statistics, Hayward, California.
Diaconis, P., and Shahshahani, M. (1981). Generating a random permutation with random transpositions. Z. Wahrsch. Verw. Gebiete 57, 159-179.
Diaconis, P., and Shahshahani, M. (1987). Time to reach stationarity in the Bernoulli-Laplace diffusion model. SIAM J. Math. Anal. 18, 208-218.
Greenhalgh, A. S. (1989). Random walks on groups with subgroup invariance properties. Technical report No. 321, Department of Statistics, Stanford University.
James, G., and Kerber, A. (1981). The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and Its Applications, Vol. 16, Addison-Wesley, Reading, Massachusetts.
MacDonald, I. G. (1995). Symmetric Functions and Hall Polynomials, 2nd ed., Clarendon, Oxford, England.
Scarabotti, F. (1997). Time to reach stationarity in the Bernoulli-Laplace diffusion model with many urns. Adv. Appl. Math. 18, 351-371.
Schoolfield, C. H. (1998). Random walks on wreath products of groups and Markov chains on related homogeneous spaces. Ph.D. dissertation, Department of Mathematical Sciences, The Johns Hopkins University.
Schoolfield, C. H. (2002). Random walks on wreath products of groups. J. Theoret. Prob. To appear.
Serre, J.-P. (1977). Linear Representations of Finite Groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York.
Tokuyama, T. (1984). On the decomposition rules of tensor products of the representations of the classical Weyl groups. J. Algebra 88, 380-394.
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Schoolfield, C.H. A Signed Generalization of the Bernoulli–Laplace Diffusion Model. Journal of Theoretical Probability 15, 97–127 (2002). https://doi.org/10.1023/A:1013841306577
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DOI: https://doi.org/10.1023/A:1013841306577