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Funktsional. Anal. i Prilozhen., 2013, Volume 47, Issue 2, Pages 68–79 (Mi faa3106)  

This article is cited in 2 scientific papers (total in 2 papers)

Asymptotics of Products of Nonnegative Random Matrices

V. Yu. Protasov

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

Abstract: Asymptotic properties of products of random matrices $\xi_k=X_k\cdots X_1$ as $k\to\infty$ are analyzed. All product terms $X_i$ are independent and identically distributed on a finite set of nonnegative matrices $\mathcal{A}=\{A_1,…, A_m\}$. We prove that if $\mathcal{A}$ is irreducible, then all nonzero entries of the matrix $\xi_k$ almost surely have the same asymptotic growth exponent as $k\to\infty$, which is equal to the largest Lyapunov exponent $\lambda(\mathcal{A})$. This generalizes previously known results on products of nonnegative random matrices. In particular, this removes all additional “nonsparsity” assumptions on matrices imposed in the literature. We also extend this result to reducible families. As a corollary, we prove that Cohen's conjecture (on the asymptotics of the spectral radius of products of random matrices) is true in case of nonnegative matrices.

Keywords: random matrix, Lyapunov exponent, nonnegative matrix, asymptotics, sparsity, irreducibility

DOI: https://doi.org/10.4213/faa3106

Full text: PDF file (211 kB)
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English version:
Functional Analysis and Its Applications, 2013, 47:2, 138–147

Bibliographic databases:

Document Type: Article
UDC: 517.98+519.2+512.643
Received: 05.09.2012

Citation: V. Yu. Protasov, “Asymptotics of Products of Nonnegative Random Matrices”, Funktsional. Anal. i Prilozhen., 47:2 (2013), 68–79; Funct. Anal. Appl., 47:2 (2013), 138–147

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    2. V. Yu. Protasov, “The Euler binary partition function and subdivision schemes”, Math. Comp., 86:305 (2017), 1499–1524  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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