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Lyapunov exponents for some products of random matrices: Exact expressions and asymptotic distributions. (English) Zbl 0584.60018
Random matrices and their applications, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Brunswick/Maine 1984, Contemp. Math. 50, 121-141 (1986).
[For the entire collection see Zbl 0581.00014.]
Consider the Lyapunov exponents $$\mu_ 1^ N\geq...\geq \mu^ N_ N$$ for products of i.i.d. $$N\times N$$ matrices, A(1),A(2),.... If the $$N^ 2$$ entries of A(1) are i.i.d. normal with zero mean and variance 1/N, then for each k, $$\mu^ N_ k=[\log (2/N)+\Psi ((N-k+1)/2)]/2$$ where $$\Psi =\Gamma '/\Gamma$$ is the digamma function. In this case, the empirical distribution for $$\{\lambda^ N_ k:=\exp (\mu^ N_ k)\}$$ converges as $$N\to \infty$$ to H($$\lambda)$$, a ”triangle law”, whereas the asymptotic distribution, K, for the (random) eigenvalues of $$| A(1)|:=(A(1)^ TA(1))^{1/2}$$ is Wigner’s ”quarter-circle” law.
More generally, when $$| A(1)|$$ has a rotationally invariant distribution and a nonrandom limiting K, one can show, under appropriate assumptions, that there is a limiting H($$\lambda)$$ satisfying $\int t^ 2[H\lambda^ 2+(1-H)t^ 2]^{-1}dK(t)=1.$ When the entries, $$(A(1))_{ij}$$, are independent normal with zero mean and variance $$\sigma^ 2_ i/N$$ and the $$\sigma_ i's$$ have a limiting distribution $$\tilde K,$$ one can show, under appropriate assumptions, that there is a limiting H($$\lambda)$$ satisfying $\int t^ 2[\lambda^ 2+(1-H)t^ 2]^{-1}d\tilde K(t)=1.$ These identities relating H to K or $$\tilde K$$ are expected to be valid under much more general hypotheses on the distribution of A(1) than assumed here.

##### MSC:
 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization 60F99 Limit theorems in probability theory 60G99 Stochastic processes 28D99 Measure-theoretic ergodic theory