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. . ( ~ ) P ( X , Y )= T ( Y ) ~ ( Y , x) . By symmetry, P has eigenvalues 1= Po > P, 2 2 Pi,,-, 2 -1.This paper develops methods for getting upper and lower bounds on Pi by comparison with a second reversible chain on the same state space. This extends the ideas introduced in Diaconis and Saloff-Coste (1993), where random walks on finite groups were… (More)
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/about/terms.html. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in… (More)
The Metropolis algorithm is a widely used procedure for sampling from a specified distribution on a large finite set. We survey what is rigorously known about running times. This includes work from statistical physics, computer science, probability and statistics. Some new results are given ae an illustration of the geometric theory of Markov chains.
Markov chains on finite sets are used in a great variety of situations to approximate, understand and sample from their limit distribution. A familiar example is provided by card shuffling methods. From this viewpoint, one is interested in the “mixing time” of the chain, that is, the time at which the chain gives a good approximation of the limit… (More)
Given a Riemannian manifold (M, g), we study the solutions of heat equations associated with second order differential operators in divergence form that are uniformly elliptic with respect to g . Typical examples of such operators are the Laplace operators of Riemannian structures which are quasi-isometric to g . We first prove some Poincare and Sobolev… (More)
We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.
We consider the problem of giving explicit spectral bounds for time inhomogeneous Markov chains on a finite state space. We give bounds that apply when there exists a probability π such that each of the different steps corresponds to a nice ergodic Markov kernel with stationary measure π . For instance, our results provide sharp bounds for models such as… (More)
This paper gives a necessary and sufficient condition for a sequence of birth and death chains to converge abruptly to stationarity, that is, to present a cut-off. The condition involves the notions of spectral gap and mixing time. Y. Peres has observed that for many families of Markov chains, there is a cutoff if and only if the product of spectral gap and… (More)
Wreath products are a type of semidirect product. They play an important role in group theory. This paper studies the basic behavior of simple random walks on such groups and shows that these walks have interesting, somewhat exotic behaviors. The crucial fact is that the probability of return to the starting point of certain walks on wreath products is… (More)