Bulk behaviour of Schur-Hadamard products of symmetric random matrices

We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur-Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSDs of Schur-Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur-Hadamard product of independent Toeplitz and Hankel matrices has the semi-circular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.Comment: 27 pages, 1 figure; to appear, Random Matrices: Theory and Applications. This is the final version, incorporating referee comment

Some characterisation results on classical and free Poisson thinning

Poisson thinning is an elementary result in probability, which is of great importance in the theory of Poisson point processes. In this article, we record a couple of characterisation results on Poisson thinning. We also consider free Poisson thinning, the free probability analogue of Poisson thinning, which arises naturally as a high-dimensional asymptotic analogue of Cochran's theorem from multivariate statistics on the "Wishart-ness" of quadratic functions of Gaussian random matrices. The main difference between classical and free Poisson thinning is that, in the former, the involved Poisson random variable can have an arbitrary mean, whereas, in the free version, the "mean" of the relevant free Poisson variable must be 1. We prove similar characterisation results for free Poisson thinning and note their implications in the context of Cochran's theorem.Comment: 19 pages, 1 figure. Added the free analogue of Craig's theorem in this version and streamlined some proof

Bulk behaviour of skew-symmetric patterned random matrices

Limiting Spectral Distributions (LSD) of real symmetric patterned matrices have been well-studied. In this article, we consider skew-symmetric/anti-symmetric patterned random matrices and establish the LSDs of several common matrices. For the skew-symmetric Wigner, skew-symmetric Toeplitz and the skew-symmetric Circulant, the LSDs (on the imaginary axis) are the same as those in the symmetric cases. For the skew-symmetric Hankel and the skew-symmetric Reverse Circulant however, we obtain new LSDs. We also show the existence of the LSDs for the triangular versions of these matrices. We then introduce a related modification of the symmetric matrices by changing the sign of the lower triangle part of the matrices. In this case, the modified Wigner, modified Hankel and the modified Reverse Circulants have the same LSDs as their usual symmetric counterparts while new LSDs are obtained for the modified Toeplitz and the modified Symmetric Circulant.Comment: 21 pages, 2 figure

Consistent model selection in the spiked Wigner model via AIC-type criteria

Consider the spiked Wigner model $$X = \sum_{i = 1}^k \lambda_i u_i u_i^\top + \sigma G,$$ where $G$ is an $N \times N$ GOE random matrix, and the eigenvalues $\lambda_i$ are all spiked, i.e. above the Baik-Ben Arous-P\'ech\'e (BBP) threshold $\sigma$. We consider AIC-type model selection criteria of the form $$-2 \, (\text{maximised log-likelihood}) + \gamma \, (\text{number of parameters})$$ for estimating the number $k$ of spikes. For $\gamma > 2$, the above criterion is strongly consistent provided $\lambda_k > \lambda_{\gamma}$, where $\lambda_{\gamma}$ is a threshold strictly above the BBP threshold, whereas for $\gamma < 2$, it almost surely overestimates $k$. Although AIC (which corresponds to $\gamma = 2$) is not strongly consistent, we show that taking $\gamma = 2 + \delta_N$, where $\delta_N \to 0$ and $\delta_N \gg N^{-2/3}$, results in a weakly consistent estimator of $k$. We also show that a certain soft minimiser of AIC is strongly consistent.Comment: 14 pages, 1 figure, 3 table

∗*-convergence of Schur-Hadamard products of independent non-symmetric random matrices

Let $\{x_{\alpha}\}_{\alpha\in\mathbb{Z}}$ and $\{y_{\alpha}\}_{\alpha\in\mathbb{Z}}$ be two independent sets of zero mean, unit variance random variables with uniformly bounded moments of all orders. Consider a non-symmetric 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. We show that $n^{-1/2}M_n$, as an element of the $*$-probability space $(\mathcal{M}_n(L^{\infty,-}(\Omega,\mathbb{P})), \frac{1}{n}\mathbb{E}\mathrm{tr})$, converges in $*$-distribution to a circular variable. This gives a matrix model for circular variables with only $O(n)$ bits of randomness. As a direct corollary, we recover a result of Bose and Mukherjee (2014) that the empirical spectral measure of the $n^{-1/2}$-scaled Schur-Hadamard product of symmetric Toeplitz and Hankel matrices converges weakly almost surely to the semi-circular law. Based on numerical evidence, we conjecture that the circular law $\mu_{\mathrm{circ}}$, i.e. the uniform measure on the unit disk of $\mathbb{C}$, also the Brown measure of $n^{-1/2}M_n$, is in fact the limiting spectral measure of $n^{-1/2}M_n$. If true, this would be an interesting example where a random matrix with only $O(n)$ bits of randomness has the circular law as its limiting spectral measure (all the standard examples have $\Omega(n^2)$ bits of randomness). More generally, we prove similar results for structured random matrices of the form $A=((a_{L(i,j)}))$ with link function $L:\mathbb{Z}_+^2\rightarrow\mathbb{Z}^d$. Given two such ensembles of matrices with link functions $L_X$ and $L_Y$, we show that $*$-convergence to a circular variable holds for their Schur-Hadamard product if the map $$(i,j) \mapsto (L_X(i,j), L_Y(i,j))$$ is injective and some mild regularity assumptions on $L_X$ and $L_Y$ are satisfied.Comment: 17 pages, 3 figure