Scaling positive random matrices: concentration and asymptotic convergence

Non-Asymptotic Theory of Random Matrices

1 Definition The topic in this lecture is Subgaussian random variables. We start with the definition, and discuss some properties they hold. Definition 1 (Subgaussian random variables). A random variable X is subgaussian if ∃c, C such that P(|x| > t) ≤ Ce −ct 2 ∀t ≥ 0. (1) As the name suggests, the notion of subgaussian random variables is a generalization of Gaussian random variables. Both the...

Asymptotic Lyapunov Exponents for Large Random Matrices

Suppose that A1, . . . , AN are independent random matrices whose atoms are iid copies of a random variable ξ of mean zero and variance one. It is known from the works of Newman et. al. in the late 80s that when ξ is gaussian then N−1 log ‖AN . . . A1‖ converges to a non-random limit. We extend this result to more general matrices with explicit rate of convergence. Our method relies on a simple...

Asymptotic Freeness Almost Everywhere for Random Matrices

Voiculescu’s asymptotic freeness result for random matrices is improved to the sense of almost everywhere convergence. The asymptotic freeness almost everywhere is first shown for standard unitary matrices based on the computation of multiple moments of their entries, and then it is shown for rather general unitarily invariant selfadjoint random matrices (in particular, standard selfadjoint Gau...

Non-Asymptotic Theory of Random Matrices

Non-Asymptotic Theory of Random Matrices

We will estimate the size of a random process (X t) t∈T by the size of another (simpler) random process (Y t) t∈T. Define a Gaussian centered process to be when X t is a mean-zero random variable. For a random variable X