Geodesic
Invariant Functionals for Random Matrices
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 3, pp. 84-88

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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 $\mathbb{R}^d_+$. 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.
Mots-clés : random matrices
Keywords: Lyapunov exponents, invariant functions, concave homogeneous functionals, fixed point, asymptotics. 
@article{FAA_2010_44_3_a10,
     author = {V. Yu. Protasov},
     title = {Invariant {Functionals} for {Random} {Matrices}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {84--88},
     publisher = {mathdoc},
     volume = {44},
     number = {3},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2010_44_3_a10/}
}
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V. Yu. Protasov. Invariant Functionals for Random Matrices. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 3, pp. 84-88. http://geodesic.mathdoc.fr/item/FAA_2010_44_3_a10/