Commuting birth and death processes

S. Evans, B. Sturmfels, C. Uhler, The Annals of Applied Probability 20 (2010) 238–266.


Journal Article | Published
Author
; ;
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.
Publishing Year
Date Published
2010-01-01
Journal Title
The Annals of Applied Probability
Acknowledgement
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.
Volume
20
Page
238 - 266
IST-REx-ID

Cite this

Evans S, Sturmfels B, Uhler C. Commuting birth and death processes. The Annals of Applied Probability. 2010;20:238-266. doi:10.1214/09-AAP615
Evans, S., Sturmfels, B., & Uhler, C. (2010). Commuting birth and death processes. The Annals of Applied Probability, 20, 238–266. https://doi.org/10.1214/09-AAP615
Evans, Steven, Bernd Sturmfels, and Caroline Uhler. “Commuting Birth and Death Processes.” The Annals of Applied Probability 20 (2010): 238–66. https://doi.org/10.1214/09-AAP615.
S. Evans, B. Sturmfels, and C. Uhler, “Commuting birth and death processes,” The Annals of Applied Probability, vol. 20, pp. 238–266, 2010.
Evans S, Sturmfels B, Uhler C. 2010. Commuting birth and death processes. The Annals of Applied Probability. 20, 238–266.
Evans, Steven, et al. “Commuting Birth and Death Processes.” The Annals of Applied Probability, vol. 20, Institute of Mathematical Statistics, 2010, pp. 238–66, doi:10.1214/09-AAP615.

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