Algebraic criteria for ergodicity of arbitrary matrices Applications to finite Markov chains Marius...

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

Introduction

10ème Colloque Franco-Roumain de Mathématiques Appliquées. 26-31 Août 2010. Poitiers, France 2

Preliminaries



Preliminaries

10ème Colloque Franco-Roumain de Mathématiques Appliquées. 26-31 Août 2010. Poitiers, France

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Preliminaries


Preliminaries


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Algebraic criteria that a matrix be uniformly power convergent



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• 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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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


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 !

Algebraic criteria for ergodicity of arbitrary matrices Applications to finite Markov chains Marius...

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