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

This article is cited in 5 scientific papers (total in 5 papers)

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

Full text: PDF file (682 kB)
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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
Received: 11.11.2009

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

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    Citing articles on Google Scholar: Russian citations, English citations
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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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    3. V. Yu. Protasov, “Asymptotics of Products of Nonnegative Random Matrices”, Funct. Anal. Appl., 47:2 (2013), 138–147  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. A. Voynov, “Shortest positive products of nonnegative matrices”, Linear Algebra Appl., 439:6 (2013), 1627–1634  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  mathscinet  isi  elib  elib
  • Математический сборник Sbornik: Mathematics (from 1967)
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