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Unformatted text preview: Submitted to the Annals of Applied Probability RATES OF CONVERGENCE OF SOME MULTIVARIATE MARKOV CHAINS WITH POLYNOMIAL EIGENFUNCTIONS By Kshitij Khare and Hua Zhou Stanford University We provide a sharp non-asymptotic analysis of the rates of con- vergence for some standard multivariate Markov chains using spectral techniques. All chains under consideration have multivariate orthogo- nal polynomial as eigenfunctions. Our examples include the Moran’s model in population genetics and its variants in community ecol- ogy, the Dirichlet-Multinomial Gibbs sampler, a class of generalized Bernoulli-Laplace processes, a generalized Ehrenfest urn model, and the multivariate normal autoregressive process. 1. Introduction. The theory of Markov chains is one of the most useful tools of applied probability and has numerous applications. Markov chains are used for modeling physical processes and evolution of a population in population genetics and community ecology. Another important use is sim- ulating from an intractable probability distribution. It is a well known fact that under mild conditions discussed in [ 2 ], a Markov chain converges to its stationary distribution. In the applications mentioned above, often it is useful to know exactly how many steps it takes for a Markov chain to be reasonably close to its stationary distribution. Answering this question as accurately as possible, is what finding ‘rates of convergence’ of Markov chains is about. In the current paper, we provide a sharp non-asymptotic analysis of rates of convergence to stationarity for a variety of multivariate Markov chains. This helps determine exactly what number of steps is necessary and suffi- cient for convergence. These Markov chains appear as standard models in population genetics, ecology, statistics, and image processing. Here is an example of our results. In community ecology, scientists study diversity and species abundances in ecological communities. The Unified Neutral Theory of Biodiversity and Biogeography (UNTB) is an important theory proposed by ecologist Stephen Hubbell in his monograph [ 26 ]. There are two levels in Hubbell’s theory, a metacommunity and a local community. AMS 2000 subject classifications: Primary 60J10; secondary 60J22,33C50 Keywords and phrases: convergence rate, Markov chains, multivariate orthogonal poly- nomials 1 imsart-aap ver. 2007/12/10 file: MultiConvRate-AAP.tex date: June 18, 2008 2 The metacommunity has constant population size N M and evolves as fol- lows. At each step, a randomly chosen individual is replaced by a new one. With probability s (speciation), the new individual is a new species that never occurs before. With probability 1 − s (no speciation), the new individ- ual is a copy of one (randomly chosen) of the remaining N M − 1 individuals....
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