Random matrices: Law of the iterated logarithm

TL;DR

This paper explores the possibility of establishing a law of the iterated logarithm for various matrix parameters, demonstrating it for the log of the permanent of Bernoulli matrices and discussing open questions.

Contribution

It proves the law of the iterated logarithm for the log of the permanent of Bernoulli matrices and raises open questions for other matrix statistics.

Findings

Law of the iterated logarithm holds for log-permanent of Bernoulli matrices

Central limit theorems are established for eigenvalues and log-determinant

Open problems are posed for other matrix parameters

Abstract

The theory of random matrices contains many central limit theorems. We have central limit theorems for eigenvalues statistics, for the log-determinant and log-permanent, for limiting distribution of individual eigenvalues in the bulk, and many others. In this notes, we discuss the following problem: Is it possible to prove the law of the iterated logarithm? We illustrate this possibility by showing that this is indeed the case for the log of the permanent of random Bernoulli matrices and pose open questions concerning several other matrix parameters.