## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 27, Number 6 (2017), 3672-3705.

### Asymptotic Lyapunov exponents for large random matrices

#### Abstract

Suppose that $A_{1},\dots ,A_{N}$ 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 \Vert A_{N}\dots A_{1}\Vert $ converges to $\log \sqrt{n}$ as $N\to \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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 27, Number 6 (2017), 3672-3705.

**Dates**

Received: July 2016

Revised: January 2017

First available in Project Euclid: 15 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1513328711

**Digital Object Identifier**

doi:10.1214/17-AAP1293

**Mathematical Reviews number (MathSciNet)**

MR3737935

**Zentralblatt MATH identifier**

06848276

**Subjects**

Primary: 15A52 60B10: Convergence of probability measures

**Keywords**

Lyapunov exponents large random matrices

#### Citation

Nguyen, Hoi H. Asymptotic Lyapunov exponents for large random matrices. Ann. Appl. Probab. 27 (2017), no. 6, 3672--3705. doi:10.1214/17-AAP1293. https://projecteuclid.org/euclid.aoap/1513328711