RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Subscription License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

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

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)
References: PDF file   HTML file

English version:
Functional Analysis and Its Applications, 2013, 47:2, 138–147

Bibliographic databases:

UDC: 517.98+519.2+512.643

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

Citation in format AMSBIB
\Bibitem{Pro13} \by V.~Yu.~Protasov \paper Asymptotics of Products of Nonnegative Random Matrices \jour Funktsional. Anal. i Prilozhen. \yr 2013 \vol 47 \issue 2 \pages 68--79 \mathnet{http://mi.mathnet.ru/faa3106} \crossref{https://doi.org/10.4213/faa3106} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3113870} \zmath{https://zbmath.org/?q=an:06207381} \elib{http://elibrary.ru/item.asp?id=20730691} \transl \jour Funct. Anal. Appl. \yr 2013 \vol 47 \issue 2 \pages 138--147 \crossref{https://doi.org/10.1007/s10688-013-0018-8} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000321438400006} \elib{http://elibrary.ru/item.asp?id=20439337} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84879822947} 

• http://mi.mathnet.ru/eng/faa3106
• https://doi.org/10.4213/faa3106
• http://mi.mathnet.ru/eng/faa/v47/i2/p68

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. L. Speidel, K. Klemm, V. M. Eguiluz, N. Masuda, “Temporal interactions facilitate endemicity in the susceptible-infected-susceptible epidemic model”, New J. Phys., 18 (2016), 073013
2. V. Yu. Protasov, “The Euler binary partition function and subdivision schemes”, Math. Comp., 86:305 (2017), 1499–1524
•  Number of views: This page: 389 Full text: 104 References: 47 First page: 35