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 nonasymptotic 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 DirichletMultinomial Gibbs sampler, a class of generalized
BernoulliLaplace 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 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 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
imsartaap ver. 2007/12/10 file: MultiConvRateAAP.tex date: June 18, 2008
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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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 Fall '08
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 Markov Chains, Probability, Orthogonal polynomials, imsartaap ver, MultiConvRateAAP.tex date

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