The Annals of Applied Probability

Commuting birth-and-death processes

Steven N. Evans, Bernd Sturmfels, and Caroline Uhler

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We use methods from combinatorics and algebraic statistics to study analogues of birth-and-death processes that have as their state space a finite subset of the m-dimensional lattice and for which the m matrices that record the transition probabilities in each of the lattice directions commute pairwise. One reason such processes are of interest is that the transition matrix is straightforward to diagonalize, and hence it is easy to compute n step transition probabilities. The set of commuting birth-and-death processes decomposes as a union of toric varieties, with the main component being the closure of all processes whose nearest neighbor transition probabilities are positive. We exhibit an explicit monomial parametrization for this main component, and we explore the boundary components using primary decomposition.

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Ann. Appl. Probab., Volume 20, Number 1 (2010), 238-266.

First available in Project Euclid: 8 January 2010

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Primary: 60J22: Computational methods in Markov chains [See also 65C40] 60C05: Combinatorial probability 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 68W30: Symbolic computation and algebraic computation [See also 11Yxx, 12Y05, 13Pxx, 14Qxx, 16Z05, 17-08, 33F10]

Birth-and-death process regime switching reversible orthogonal polynomial binomial ideal toric commuting variety Markov basis Graver basis unimodular matrix matroid primary decomposition


Evans, Steven N.; Sturmfels, Bernd; Uhler, Caroline. Commuting birth-and-death processes. Ann. Appl. Probab. 20 (2010), no. 1, 238--266. doi:10.1214/09-AAP615.

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