Stationary distribution of product of matrices with random coefficients

Authors

  • E.A. Illarionov Lomonosov Moscow State University
  • V.N. Tutubalin Lomonosov Moscow State University
  • D.D. Sokoloff Lomonosov Moscow State University

Keywords:

stationary distribution, product of matrices, integral equation, Jacobi equation

Abstract

The study devoted to the probabilistic product properties for a large number of independent equally distributed random matrices is based on a number of results obtained by H. Furstenberg (1963). Particularly, he proved the ergodicity of Markov chains caused by the action of random matrices on some compact uniform subspace of the group of matrices W called the boundary of this group. The stationary distribution of this chain (the invariant probability measure) defines the characteristics of the limiting behavior of the matrix product. Up to now, this measure was found only in simple cases. As an example, we consider the fundamental matrix of the Jacobi equation with random curvature to compute the invariant measure. Using this measure, we compute the Lyapunov exponent and the growth rate of statistical moments of the Jacobi field. Our results are compared with the results obtained previously by the application of the Monte Carlo method; a high degree of coincidence is observed.

Author Biographies

E.A. Illarionov

V.N. Tutubalin

D.D. Sokoloff

References

  1. Bellman R. Limit theorem for non-commutative operations // I. Duke Math. J. 1954. 21. 491-500.
  2. Bougerol P., Lacroix J. Product of random matrices with application to Schrödinger operators // Progress in Probability and Statistics. 1985. 8. 1-283.
  3. Comtet A., Texier C., Tourigny Y. Products of random matrices and generalized quantum point scatterers // J. of Statistical Physics. 2010. 140. 427-466.
  4. Furstenberg H. Noncommuting random products // Trans. Amer. Math. Soc. 1963. 108. 377-428.
  5. Tutubalin V.N. A central limit theorem for products of random matrices and some of its applications // Symposia Mathematica. 1977. XXI. 101-116.
  6. Михайлов Е.А., Соколов Д.Д., Тутубалин В.Н. Фундаментальная матрица для уравнений Якоби со случайными коэффициентами // Вычислительные методы и программирование. 2010. 11. 103-110.
  7. Zeldovich Ya.B., Ruzmaikin A.A., Molchanov S.A., Sokoloff D.D. Kinematic dynamo problem in a linear velocity field // J. Fluid Mech. 1984. 144. 1-11.

Published

24-04-2012

How to Cite

Илларионов Е., Тутубалин В., Соколов Д. Stationary Distribution of Product of Matrices With Random Coefficients // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2012. 13. 218-225

Issue

Section

Section 1. Numerical methods and applications