algebraic criteria for ergodicity of arbitrary matrices applications to finite markov chains marius...
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Algebraic criteria for ergodicity of arbitrary matricesApplications to finite Markov chains
Marius Radulescu and Sorin RadulescuInstitute of Mathematical Statistics and Applied Mathematics “Gheorghe Mihoc - Caius Iacob” , Calea 13 Septembrie nr. 13,
Bucharest 5, RO-050711, ROMANIAe-mail:[email protected]
110ème Colloque Franco-Roumain de Mathématiques Appliquées.
26-31 Août 2010. Poitiers, France
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Introduction
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Preliminaries
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10ème Colloque Franco-Roumain de Mathématiques Appliquées.
Preliminaries
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Preliminaries
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Preliminaries
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Algebraic criteria that a matrix be uniformly power convergent
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Algebraic criteria that a matrix be uniformly power convergent
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Algebraic criteria that a matrix be uniformly power convergent
• References• K. HENSEL (1926), ber Potenzreihen yon Matrizen, J. Reine Angew. Math., 155,
pp. 107-110.• R. OLDENBURGER (1940), Infinite powers o] matrices and characteristic roots,
Duke Math. J., 6, pp. 357-361. Samuel Karlin and Howard. M. Taylor, A Second Course in Stochastic Processes- (New York: Academic Press, 1981, xviii + 542
• A.N. Shiryayev. Probability, volume 95 of GTM. Springer, 1984.• M. Cardona, M.A. Colomer, J. Conde, J. Miret, J. Miró, A. Zaragoza. Markov
chains: computing limit existence and approximations with DNA. Biosystems, Volum 81, pp. 411-427, 2005.
• Carl D. Meyer, Jr. and R. J. Plemmons, Convergent Powers of a Matrix with Applications to Iterative Methods for Singular Linear Systems, SIAM Journal on Numerical Analysis, Vol. 14, No. 4 (Sep., 1977), pp. 699-705.
• Stephen H. Friedberg, Arnold J. Insel, Convergence of matrix powers, International Journal of Mathematical Education in Science and Technology, Volume 23, Issue 5 September 1992 , pages 765 – 769.
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Algebraic criteria that a matrix be uniformly power convergent
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Algebraic criteria that a matrix be uniformly power convergent
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Limit of the powers of matrices
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Limit of the powers of matrices
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Limit of the powers of matrices
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Limit of the powers of matrices
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Limit of the powers of matrices
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The structure theorem of stochastic matrices
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Algebraic criteria that a stochastic matrix be uniformly ergodic
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Algebraic criteria that a stochastic matrix be uniformly ergodic
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Applications to Markov chains
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A formula for the limiting matrix
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Formulation of new problems
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The End
• Thank you very much for your attention !