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 Mat. Sb., 2011, Volume 202, Number 1, Pages 105–132 (Mi msb7651)

Invariant functions for the Lyapunov exponents of random matrices

V. Yu. Protasov

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: A new approach to the study of Lyapunov exponents of random matrices is presented. We prove that any family of nonnegative $(d\times d)$-matrices has a continuous concave invariant functional on $\mathbb R^d_+$. Under some standard assumptions on the matrices, this functional is strictly positive, and the coefficient corresponding to it is equal to the largest Lyapunov exponent. As a corollary we obtain asymptotics for the expected value of the logarithm of norms of matrix products and of their spectral radii. Another corollary gives new upper and lower bounds for the Lyapunov exponent, and an algorithm for computing it for families of nonnegative matrices. We consider possible extensions of our results to general nonnegative matrix families and present several applications and examples.
Bibliography: 29 titles.

Keywords: random matrices, Lyapunov exponents, invariant functions, concave homogeneous functionals, fixed point, asymptotics.

DOI: https://doi.org/10.4213/sm7651

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English version:
Sbornik: Mathematics, 2011, 202:1, 101–126

Bibliographic databases:

Document Type: Article
UDC: 517.98+519.2
MSC: Primary 60H25; Secondary 15A60

Citation: V. Yu. Protasov, “Invariant functions for the Lyapunov exponents of random matrices”, Mat. Sb., 202:1 (2011), 105–132; Sb. Math., 202:1 (2011), 101–126

Citation in format AMSBIB
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• https://doi.org/10.4213/sm7651
• http://mi.mathnet.ru/eng/msb/v202/i1/p105

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This publication is cited in the following articles:
1. Protasov V.Yu., Voynov A.S., “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765
2. Protasov V.Yu., Jungers R.M., “Lower and upper bounds for the largest Lyapunov exponent of matrices”, Linear Algebra Appl., 438:11 (2013), 4448–4468
3. V. Yu. Protasov, “Asymptotics of Products of Nonnegative Random Matrices”, Funct. Anal. Appl., 47:2 (2013), 138–147
4. A. Voynov, “Shortest positive products of nonnegative matrices”, Linear Algebra Appl., 439:6 (2013), 1627–1634
5. V. Zh. Sakbaev, “On the law of large numbers for compositions of independent random semigroups”, Russian Math. (Iz. VUZ), 60:10 (2016), 72–76
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