Commuting birth and death processes Journal Article

Author(s): Evans, Steven N; Sturmfels, Bernd; Uhler, Caroline
Article Title: Commuting birth and death processes
Abstract: 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.
Keywords: Markov basis; Binomial ideal; Birth-and-death process; Commuting variety; Graver basis; Matroid; Orthogonal polynomial; Primary decomposition; Regime switching; Reversible; Toric; Unimodular matrix
Journal Title: The Annals of Applied Probability
Volume: 20
ISSN: 1050-5164
Publisher: Institute of Mathematical Statistics  
Date Published: 2010-01-01
Start Page: 238
End Page: 266
DOI: 10.1214/09-AAP615
Notes: Steven N. Evans was supported in part by NSF Grants DMS-04-05778 and DMS-09-07630. Bernd Sturmfels was supported in part by NSF Grants DMS-04-56960 and DMS-07-57236. Caroline Uhler was supported by an International Fulbright Science and Technology Fellowship.
Open access: yes (repository)
IST Austria Authors
  1. Caroline Uhler
    26 Uhler
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