Abstract
The discrete-time phase type (PH) distributions are used in the numerical solution of many problems. The representation of a PH distribution consists of a vector and a substochastic matrix. The common feature of the PH distributions is that they have rational moment generating function. The moment generating function depends on the eigendecomposition of the transition probability matrix of the PH distribution, but the eigenvalues and eigenvectors are important in other cases as well. Due to the finite precision of the numerical calculations or other reasons, a representation may contain errors that change the eigendecomposition. This paper presents upper bounds on the change of the eigendecomposition when the PH representation is perturbed. Since these bounds do not depend on a particular representation, but they hold for all, the analysed PH representations have uniformly stable spectral decompositions.
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References
G. E. Cho and C. D. Meyer, Comparison of Perturbation Bounds for the Stationary Distribution of a Markov Chain, Linear Algebra Appl., 335 (2001), 137–150.
J. W. Demmel, Computing stable eigendecompositions of matrices, Linear Algebra Appl., 79 (1986), 163–193.
G. Latouche and V. Ramaswami, Introduction to Matrix-Analytic Methods in Stochastic Modeling, Series on statistics and applied probability, ASA-SIAM, 1999.
G. W. Stewart, Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIAM Rev., 15 (1973), 727–764.
G. W. Stewart and J. Sun, Matrix Perturbation Theory, Academic Press, San Diego, 1990.
L. N. Trefethen and M. Embree, Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators, Princeton University Press, 2005.
J. H. Wilkinson, Sensitivity of eigenvalues II, Utilitas Math., 30 (1986), 243–286.
J. H. Wilkinson, On a theorem of Feingold, Linear Algebra Appl., 88/89 (1987), 13–30.
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Communicated by Dénes Petz
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Éltető, T. Why certain discrete phase type representations have numerically stable spectral decomposition. Period Math Hung 54, 107–120 (2007). https://doi.org/10.1007/s-10998-007-1107-3
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DOI: https://doi.org/10.1007/s-10998-007-1107-3