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Asymptotic Lyapunov exponents for large random matrices. (English) Zbl 1386.60027

Summary: Suppose that \(A_{1},\ldots ,A_{N}\) are independent random matrices of size \(n\) whose entries are i.i.d. copies of a random variable \(\xi \) of mean zero and variance one. It is known from the late 1980s that when \(\xi \) is Gaussian then \(N^{-1}\log \| A_{N}\ldots A_{1}\| \) converges to \(\log \sqrt{n}\) as \(N\rightarrow \infty \). We will establish similar results for more general matrices with explicit rate of convergence. Our method relies on a simple interplay between additive structures and growth of matrices.

MSC:

60B20 Random matrices (probabilistic aspects)
60B10 Convergence of probability measures
15B52 Random matrices (algebraic aspects)
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