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Stability of Exponential Rate of Growth of Products of Random Matrices Under Local Random Perturbations
Author(s) -
Slud Eric V.
Publication year - 1986
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-33.1.180
Subject(s) - mathematics , sequence (biology) , independent and identically distributed random variables , random matrix , combinatorics , random variable , matrix (chemical analysis) , limit (mathematics) , stability (learning theory) , circular law , discrete mathematics , mathematical analysis , statistics , physics , sum of normally distributed random variables , eigenvalues and eigenvectors , genetics , materials science , quantum mechanics , machine learning , computer science , composite material , biology
It is known that for any sequence X 1 X 2 , … of identically distributed independent random matrices with a common distribution μ, the limitΛ ( μ ) =lim n → ∞ = n − 1 log ‖ X n … X 1 ‖exists with probability 1. We show that if μ has compact support in GL( m , R) and if for k ⩾ 1 { X i ( k )} i are ‘locally perturbed’ i.i.d. sequences which have laws μ k , and which satisfy ∥ X ( k ) i f X 1 ⩾1/ k almost surely and a further technical condition, then Δ(μ k )→ k →∞ as Extensions are proved for Markovian matrix sequences and for some nonlinearly perturbed matrix sequences.