On ∗-Convergence of Schur-Hadamard Products of Independent Nonsymmetric Random Matrices

Article Type

Research Article

Publication Title

International Mathematics Research Notices

Abstract

Let and be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric Toeplitz matrix and a Hankel matrix, and let be their elementwise/Schur-Hadamard product. In this article, we show that almost surely,, as an element of the ∗-probability space, converges in ∗-distribution to a circular variable. With i.i.d. Rademacher entries, this construction gives a matrix model for circular variables with only bits of randomness. We also consider a dependent setup where and are independent strongly multiplicative systems (à la Gaposhkin [7]) satisfying an additional admissibility condition, and have uniformly bounded moments of all orders - a nontrivial example of such a system being, where. In this case, we show in-expectation and in-probability convergence of the ∗-moments of to those of a circular variable. Finally, we generalise our results to Schur-Hadamard products of structured random matrices of the form and, under certain assumptions on the link-functions and, most notably the injectivity of the map. Based on numerical evidence, we conjecture that the circular law, that is, the uniform measure on the unit disk of, which is also the Brown measure of a circular variable, is in fact the limiting spectral measure (LSM)of. If true, this would furnish an interesting example where a random matrix with only bits of randomness has the circular law as its LSM.

First Page

14667

Last Page

14698

DOI

https://10.1093/imrn/rnac215

Publication Date

8-1-2023

Comments

Open Access, Green

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