Finer filtration for matrix-valued cocycle based on Oseledec's multiplicative ergodic theorem

We consider a measurable matrix-valued cocycle A : Z+ × X → Rd×d, driven by a measurepreserving transformation T of a probability space ( X,F , μ), with the integrability condition log ‖A(1, ·)‖ ∈ L1(μ). We show that forμ-a.e.x ∈ X, if lim n→∞ 1 n log‖A(n, x)v‖ = 0 for all v ∈ Rd \ {0}, then the trajectory{A(n, x)v}n=0 is far away from 0 (i.e. lim sup n→∞ ‖A(n, x)v‖ > 0) and there is some nonzero v such that lim sup n→∞ ‖A(n, x)v‖ ≥ ‖v‖. This improves the classical multiplicative ergodic theorem of Oseledeč. We here prese nt an application to linear random processes to illustrate the importance.