A companion to the Oseledec multiplicative ergodic theorem
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- by Joseph C. Watkins
- Proc. Amer. Math. Soc. 99 (1987), 772-776
- DOI: https://doi.org/10.1090/S0002-9939-1987-0877055-7
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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 $E[{\sup _{0 \leq p \leq 1}}||{F’_k}(p)||] < \infty$. Let $\mathcal {I}$ denote the invariant sigma field for the sequence. Then \[ \lim \limits _{n \to \infty } {F_n}\left ( {\frac {1}{n}} \right ) \cdots {F_2}\left ( {\frac {1}{n}} \right ){F_1}\left ( {\frac {1}{n}} \right ) = \exp E[{F’_1}(0)|\mathcal {I}]\] with probability one.References
- V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280 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.
Bibliographic Information
- © Copyright 1987 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 99 (1987), 772-776
- MSC: Primary 60B15; Secondary 28D05
- DOI: https://doi.org/10.1090/S0002-9939-1987-0877055-7
- MathSciNet review: 877055