# Random walks and Lyapunov exponents

Given a sequence $$Y_1, Y_2, \dots$$ of i.i.d. matrices in $$\mathrm{GL}_n(\mathbb R)$$, there is a theorem of Furstenberg and Kesten which says that if $$\mathbb E(\log\|Y_1\|)$$ is finite, there exists a constant $$\gamma$$ (the Lyapunov exponent) such that $$\lim_{n\rightarrow\infty}\frac{1}{n}\log\|Y_n\dots Y_1\| = \gamma$$ There are also versions of central limit theorems for this scenario. I'm pretty sure this is also known in a more general case (e.g. suppose we have a sequence of matrices $$Y_i$$ of order 2, and I don't want to consider sequences of length $$n$$ in which $$\dots Y_i Y_i\dots$$ appears). I am wondering if anyone knows a good reference for theorems regarding Lyapunov exponents and central theorems in this case.

• But which case? Are you interested in the sequences of matrices which are not necessarily independent? Commented May 19, 2010 at 19:50
• Do you know: Products of Random Matrices With Applications to Schrodinger Operators. BOUGEROL Philippe, LACROIX Jean. It is maybe a little old, but good (it does many things in SL(2,R) but has references for the general cases). Commented May 19, 2010 at 20:05
• Andrey, I would like to know what are the necessary conditions on a sequence of matrices to have such theorems. More immediately though, I was looking for a reference regarding, say, Markovian sequences. I will look at the book you suggested. rpotrie, I do know that book. It discusses the iid case very clearly, but I was having trouble finding references to more general cases there. Commented May 19, 2010 at 20:34
• hi, elena, are u interested in some kind of product expansion structure in GL_d? thanks
– joe
Commented Jul 4, 2010 at 0:27

Random dynamical systems by Ludwig Arnold contains a thorough discussion of various multiplicative ergodic theorems (including the Furstenberg-Kesten result), but not the central limit theorems. As far as I remember, the case of stationary sequences of linear stochastic iterations is also included there.

Edit. Concerning central limit theorems for products of random matrices, a quick search yields this reference.

I had double checked, in the book "Products of Random Matrices With Applications to Schrodinger Operators" BOUGEROL Philippe, LACROIX Jean (chapter 5 of part 1) you will find Central Limit theorems even for markovian sequences.

The Furstenberg-Kesten result generalizes as much as you like after Kingman's subaditive theorem. For this stuff, I like this notes http://www.mat.puc-rio.br/~jairo/docs/trieste.pdf

I believe the paper

  Almost all cocycles over any hyperbolic system have non-vanishing Lyapunov exponents
Annals of Math. 167 (2008), 643-680.


available for free at http://www.preprint.impa.br/Shadows/SERIE_A/2005/384.html contains positivity results for the Lyapunov exponent over hyperbolic dynamical systems. Also the paper by C. Bonatti, X. Gomez-Mont, and M. Viana cited as [7] in there should be of interest.

I am not sure if they treat central limit theorems in these works.