## Analysis Learning Seminar

# Applications of Multiplicative LLN and CLT to Random Matrices

## Raji Majumdar and Anthony Sisti (University of Connecticut)

### Friday, March 2, 2018
3:30 pm

MONT 314

The Lyapunov exponent is the exponential growth rate of the operator norm of the partial products of a sequence of independent and identically distributed random matrices. It usually cannot be computed explicitly from the distribution of the matrices. Furstenberg and Kesten (1960) and Le Page (1982) found analogues to the Law of Large Numbers and Central Limit Theorem, respectively, for the norm of the partial products of a sequence of such random matrices. We use these analogues to efficiently compute the Lyapunov exponent for several random matrix models and numerically estimate the corresponding variances. For random matrices of order 2, with independent components distributed as $\xi\textrm{Bernoulli}(\frac{1}{2})$, where $\xi\in\mathbb{R}$, we obtain analytic estimates for the Lyapunov exponent in terms of a limit involving Fibonacci-like sequences.