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A companion to the Oseledec multiplicative ergodic theorem

Author: Joseph C. Watkins
Journal: Proc. Amer. Math. Soc. 99 (1987), 772-776
MSC: Primary 60B15; Secondary 28D05
MathSciNet review: 877055
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Abstract: Let $ {F_1},{F_2}, \ldots $ be a stationary sequence of continuously differentiable mappings from $ [0,1]$ into the set of $ d \times d$ matrices. Assume $ {F_k}(0) = I$ for each $ k$ and $ E[{\sup _{0 \leq p \leq 1}}\vert\vert{F'_k}(p)\vert\vert] < \infty $. Let $ \mathcal{I}$ denote the invariant sigma field for the sequence. Then

$\displaystyle \lim \limits_{n \to \infty } {F_n}\left( {\frac{1}{n}} \right) \c... ...}} \right){F_1}\left( {\frac{1}{n}} \right) = \exp E[{F'_1}(0)\vert\mathcal{I}]$

with probability one.

References [Enhancements On Off] (What's this?)

  • [1] V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
  • [2] Joseph C. Watkins, Limit theorems for products of random matarices: A comparison of two points of view, Proc. 1984 AMS Conf. on Random Matrices and their Products, 1985.

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Article copyright: © Copyright 1987 American Mathematical Society

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