Advances in Applied Probability

Composition Markov chains of multinomial type

Hua Zhou and Kenneth Lange

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Suppose that n identical particles evolve according to the same marginal Markov chain. In this setting we study chains such as the Ehrenfest chain that move a prescribed number of randomly chosen particles at each epoch. The product chain constructed by this device inherits its eigenstructure from the marginal chain. There is a further chain derived from the product chain called the composition chain that ignores particle labels and tracks the numbers of particles in the various states. The composition chain in turn inherits its eigenstructure and various properties such as reversibility from the product chain. The equilibrium distribution of the composition chain is multinomial. The current paper proves these facts in the well-known framework of state lumping and identifies the column eigenvectors of the composition chain with the multivariate Krawtchouk polynomials of Griffiths. The advantages of knowing the full spectral decomposition of the composition chain include (a) detailed estimates of the rate of convergence to equilibrium, (b) construction of martingales that allow calculation of the moments of the particle counts, and (c) explicit expressions for mean coalescence times in multi-person random walks. These possibilities are illustrated by applications to Ehrenfest chains, the Hoare and Rahman chain, Kimura's continuous-time chain for DNA evolution, a light bulb chain, and random walks on some specific graphs.

Article information

Adv. in Appl. Probab. Volume 41, Number 1 (2009), 270-291.

First available in Project Euclid: 21 April 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J22: Computational methods in Markov chains [See also 65C40] 60J25: Continuous-time Markov processes on general state spaces 60J27: Continuous-time Markov processes on discrete state spaces

Orthogonal polynomial Krawtchouk polynomial Ehrenfest chain inhomogeneous Markov chain coalescent time random walk convergence rate


Zhou, Hua; Lange, Kenneth. Composition Markov chains of multinomial type. Adv. in Appl. Probab. 41 (2009), no. 1, 270--291. doi:10.1239/aap/1240319585.

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