RATES OF CONVERGENCE OF SOME MULTIVARIATE MARKOV CHAINS WITH POLYNOMIAL EIGENFUNCTIONS

By Kshitij Khare, Hua Zhou. 2009

49 citations recorded by the source corpus.

We provide a sharp nonasymptotic analysis of the rates of convergence for some standard multivariate Markov chains using spectral techniques. All chains under consideration have multivariate orthogonal polynomial as eigenfunctions. Our examples include the Moran model in population genetics and its variants in community ecology, 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 simulating 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 nonasymptotic 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 sufficient 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

View Paper: https://www.semanticscholar.org/paper/da79d7032789e2ef02354fc9475708d9b9d7658f

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  1. Robert C. Griffiths
  2. Persi Diaconis
  3. Sourav Chatterjee
  4. Neil O'Connell
  5. James Allen Fill
  6. W. Koenig
  7. Kshitij Khare
  8. Ryan L. Murphy

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