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Invariant functionals for random matrices

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Functional Analysis and Its Applications Aims and scope

Abstract

A new approach to the study of the Lyapunov exponents of random matrices is presented. It is proved that, under general assumptions, any family of nonnegative matrices possesses a continuous concave positively homogeneous invariant functional (“antinorm”) on ℝ d+ . Moreover, the coefficient corresponding to an invariant antinorm equals the largest Lyapunov exponent. All conditions imposed on the matrices are shown to be essential. As a corollary, a sharp estimate for the asymptotics of the mathematical expectation for logarithms of norms of matrix products and of their spectral radii is derived. New upper and lower bounds for Lyapunov exponents are obtained. This leads to an algorithm for computing Lyapunov exponents. The proofs of the main results are outlined.

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Authors and Affiliations

  1. Moscow State University, Moscow, Russia

    V. Yu. Protasov

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  1. V. Yu. Protasov
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Correspondence to V. Yu. Protasov.

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Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 44, No. 3, pp. 84–88, 2010

Original Russian Text Copyright © by V. Yu. Protasov

This work was supported by RFBR grants 08-01-00208 and 10-01-00293, the program for support of young doctors of science (grant MD-2195.2008.1), and the program “Leading Scientific Schools” (grant NSh-3233.2008.1)

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Protasov, V.Y. Invariant functionals for random matrices. Funct Anal Its Appl 44, 230–233 (2010). https://doi.org/10.1007/s10688-010-0031-0

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  • DOI: https://doi.org/10.1007/s10688-010-0031-0

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