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November, 1993 Products of $2 \times 2$ Random Matrices
David Mannion
Ann. Appl. Probab. 3(4): 1189-1218 (November, 1993). DOI: 10.1214/aoap/1177005279


The notion of the shape of a triangle can be used to define the shape of a $2 \times 2$ real matrix; we find that the shape of a matrix retains just the right amount of information required for determining the main features of the asymptotic behaviour, as $n\rightarrow\infty$, of $\mathbf{G}_n\mathbf{G}_{n-1}\cdots\mathbf{G}_1$, where the $\mathbf{G}_i$ are i.i.d. copies of a $2 \times 2$ random matrix $\mathbf{G}$. An alternative formula to the Furstenberg formula is proposed for the upper Lyapounov exponent of the probability distribution of $\mathbf{G}$. We find that in some cases, using our formula, the Lyapounov exponent is more susceptible to explicit calculation.


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David Mannion. "Products of $2 \times 2$ Random Matrices." Ann. Appl. Probab. 3 (4) 1189 - 1218, November, 1993.


Published: November, 1993
First available in Project Euclid: 19 April 2007

zbMATH: 0784.60019
MathSciNet: MR1241041
Digital Object Identifier: 10.1214/aoap/1177005279

Primary: 60D05
Secondary: 60J15

Keywords: Contracting subsets of $\mathrm{Gl}(n, \mathbb{R})$ , Lyapounov exponent , Products of random matrices , shape

Rights: Copyright © 1993 Institute of Mathematical Statistics


Vol.3 • No. 4 • November, 1993
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