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- Title
On *-Convergence of Schur–Hadamard Products of Independent Nonsymmetric Random Matrices.
- Authors
Mukherjee, Soumendu Sundar
- Abstract
Let |$\{x_{\alpha }\}_{\alpha \in {\mathbb {Z}}}$| and |$\{y_{\alpha }\}_{\alpha \in {\mathbb {Z}}}$| be two independent collections of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a nonsymmetric Toeplitz matrix |$X_n = ((x_{i - j}))_{1 \le i, j \le n}$| and a Hankel matrix |$Y_n = ((y_{i + j}))_{1 \le i, j \le n}$| , and let |$M_n = X_n \odot Y_n$| be their elementwise/Schur–Hadamard product. In this article, we show that almost surely, |$n^{-1/2}M_n$| , as an element of the *-probability space |$(\mathcal {M}_n({\mathbb {C}}), \frac {1}{n}\text {tr})$| , converges in *-distribution to a circular variable. With i.i.d. Rademacher entries, this construction gives a matrix model for circular variables with only |$O(n)$| bits of randomness. We also consider a dependent setup where |$\{x_{\alpha }\}$| and |$\{y_{\beta }\}$| 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 |$\{\sqrt {2}\sin (2^n \pi U)\}_{n \in {\mathbb {Z}}_+}$| , where |$U \sim \textrm {Uniform}(0, 1)$|. In this case, we show in-expectation and in-probability convergence of the *-moments of |$n^{-1/2}M_n$| to those of a circular variable. Finally, we generalise our results to Schur–Hadamard products of structured random matrices of the form |$X_n = ((x_{L_X(i, j)}))_{1 \le i, j \le n}$| and |$Y_n = ((y_{L_Y(i, j)}))_{1 \le i, j \le n}$| , under certain assumptions on the link-functions |$L_X$| and |$L_Y$| , most notably the injectivity of the map |$(i, j) \mapsto (L_X(i, j), L_Y(i, j))$|. Based on numerical evidence, we conjecture that the circular law |$\mu _{\textrm {circ}}$| , that is, the uniform measure on the unit disk of |${\mathbb {C}}$| , which is also the Brown measure of a circular variable, is in fact the limiting spectral measure (LSM)of |$n^{-1/2}M_n$|. If true, this would furnish an interesting example where a random matrix with only |$O(n)$| bits of randomness has the circular law as its LSM.
- Subjects
NONSYMMETRIC matrices; RANDOM matrices; TOEPLITZ matrices; RANDOM variables; INDEPENDENT variables
- Publication
IMRN: International Mathematics Research Notices, 2023, Vol 2023, Issue 17, p14667
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnac215