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Products of 2 × 2 stochastic matrices with random entries

Published online by Cambridge University Press:  24 August 2016

Walter Van Assche
Walter Van Assche*
Affiliation:
Katholieke Universiteit Leuven
*
∗Postal address: Katholieke Universiteit Leuven, Departement Wiskunde, Celestijnenlaan 200B, B-3030 Heverlee, Belgium.
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Abstract

The limit of a product of independent 2 × 2 stochastic matrices is given when the entries of the first column are independent and have the same symmetric beta distribution. The rate of convergence is considered by introducing a stopping time for which asymptotics are given.

Keywords

LIMITS OF CONVOLUTIONSRATE OF CONVERGENCE
Type
Short Communications
Information
Journal of Applied Probability , Volume 23 , Issue 4 , December 1986 , pp. 1019 - 1024
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

The author is a research assistant of the Belgian National Fund for Scientific Research.

References

[1] Bellman, R. (1954) Limit theorems for non-commutative operators I. Duke Math. J. 21, 491–500.Google Scholar
[2] Chassaing, P., Letac, G. and Mora, M. (1984) Brocot sequences and random walks in SL(2, R), In Lecture Notes in Mathematics 1064, Springer-Verlag, Berlin.Google Scholar
[3] Chung, K. L. (1974) A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
[4] Derrida, B. and Hilhorst, H. J. (1983) Singular behaviour of certain infinite products of random 2×2 matrices. J. Phys. A 16, 2641–2654.Google Scholar
[5] Gradshteyn, I. S. and Ryzhik, I. M. (1980) Tables of Integrals, Series and Products. Academic Press, New York.Google Scholar
[6] Hille, E. (1962) Analytic Function Theory, Vol. 2. Ginn, New York.Google Scholar
[7] Kesten, H. (1973) Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248.Google Scholar
[8] Mukherjea, A. and Tserpes, N. A. (1976) Measures on Topological Semigroups., Lecture Notes in Mathematics 547, Springer-Verlag, Berlin.Google Scholar
[9] Rosenblatt, M. (1964) Equicontinuous Markov operators. Theory Prob. Appl. 9, 180–197.CrossRefGoogle Scholar
[10] Rosenblatt, M. (1965) Products of independent identically distributed stochastic matrices. J. Math. Anal. Appl. 11, 1–10.Google Scholar
[11] Rosenblatt, M. (1973) Morkov Processes: Structure and Asymptotic Behaviour. Springer-Verlag, Berlin.Google Scholar
[12] Sun, T.-C. (1975) Limits of convolutions of probability measures on the set of 2×2 stochastic matrices. Bull. Inst. Math. Sinica 3, 235–248.Google Scholar
[13] Ullman, J. L. (1972) On the regular behaviour of orthogonal polynomials. Proc. London Math. Soc. (3) 24, 119–148.Google Scholar