Abstract:
A new approach to the study of Lyapunov exponents of random matrices is presented. We prove that any family of nonnegative (d×d)(d×d)-matrices has a continuous concave invariant functional on
Rd+. 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.
\Bibitem{Pro11}
\by V.~Yu.~Protasov
\paper Invariant functions for the Lyapunov exponents of random matrices
\jour Sb. Math.
\yr 2011
\vol 202
\issue 1
\pages 101--126
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This publication is cited in the following 16 articles:
M. S. Makarov, V. Yu. Protasov, “Avtopolyarnye konicheskie tela i mnogogranniki”, Matem. sb., 216:3 (2025), 156–176
M. S. Makarov, “Antinormy i avtopolyarnye mnogogranniki”, Sib. matem. zhurn., 64:5 (2023), 1050–1064
M. S. Makarov, “Antinorms and Self-Polar Polyhedra”, Sib Math J, 64:5 (2023), 1200
K. Yu. Zamana, V. Zh. Sakbaev, “Kompozitsii nezavisimykh sluchainykh operatorov i svyazannye s nimi differentsialnye uravneniya”, Preprinty IPM im. M. V. Keldysha, 2022, 049, 23 pp.
R. Sh. Kalmetiev, Yu. N. Orlov, V. Zh. Sakbaev, “Chernoff iterations as an averaging method for random affine transformations”, Comput. Math. Math. Phys., 62:6 (2022), 996–1006
Matteo Della Rossa, Raphael M. Jungers, 2022 IEEE 61st Conference on Decision and Control (CDC), 2022, 1021
Protasov V.Yu., “Antinorms on Cones: Duality and Applications”, Linear Multilinear Algebra, 2021
Catalano C., Azfar U., Charlier L., Jungers R.M., “A Linear Bound on the K-Rendezvous Time For Primitive Sets of Nz Matrices”, Fundam. Inform., 180:4, SI (2021), 289–314
Catalano C., Jungers R.M., “The Synchronizing Probability Function For Primitive Sets of Matrices”, Int. J. Found. Comput. Sci., 31:6, SI (2020), 777–803
Yu. N. Orlov, V. Zh. Sakbaev, O. G. Smolyanov, “Feynman Formulas and the Law of Large Numbers for Random One-Parameter Semigroups”, Proc. Steklov Inst. Math., 306 (2019), 196–211
Umer Azfar, Costanza Catalano, Ludovic Charlier, Raphaël M. Jungers, Lecture Notes in Computer Science, 11647, Developments in Language Theory, 2019, 59
V. Zh. Sakbaev, “On the law of large numbers for compositions of independent random semigroups”, Russian Math. (Iz. VUZ), 60:10 (2016), 72–76
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
V. Yu. Protasov, “Asymptotics of Products of Nonnegative Random Matrices”, Funct. Anal. Appl., 47:2 (2013), 138–147
A. Voynov, “Shortest positive products of nonnegative matrices”, Linear Algebra Appl., 439:6 (2013), 1627–1634
Protasov V.Yu., Voynov A.S., “Sets of nonnegative matrices without positive products”, Linear Algebra Appl., 437:3 (2012), 749–765
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