关于独立非对称随机矩阵Schur-Hadamard乘积的$*$ -收敛性
On $*$-Convergence of Schur-Hadamard Products of Independent Nonsymmetric Random Matrices
摘要：
设$\{x_α\}_{α\in \mathbb{Z}}$和$\{y_α\}_{α\in \mathbb{Z}}$是两个独立的零均值、单位方差、各阶矩一致有界的随机变量集合。考虑一个非对称Toeplitz矩阵$X_n = ((x_{i - j}))_{1 \le i, j \le n}$和一个Hankel矩阵$Y_n = ((y_{i + j}))_{1 \le i, j \le n}$，并令$M_n = X_n \odot Y_n$是它们的逐元素/Schur-Hadamard乘积。在这篇文章中，我们几乎肯定，$n^{-1/2}M_n$，作为一个元素的$*$概率空间$(\mathcal{M}_n(\mathbb{C}), \frac{1}{n}\mathrm{tr})$，收敛于$*$分布的一个循环变量。带身份证。Rademacher条目，该构造给出了仅具有$O(n)$位随机性的循环变量的矩阵模型。我们还考虑了一个相关的设置，其中$\{x_α\}$和$\{y_β\}$是独立的强乘法系统（à la Gasthkin [7]），满足一个附加的\emph{admissibility}条件，并具有所有阶的一致有界矩-这样的系统的一个非平凡的例子是$\{\sqrt{2}\sin(2^n πU)\}_{n \in \mathbb{Z}_+}$，其中$U \sim \mathrm{Uniform}(0, 1)$。在这种情况下，我们表明在预期和概率收敛的$*$ -矩的$n^{-1/2}M_n$的一个圆形变量。最后，我们将我们的结果推广到Schur-Hadamard产品的结构随机矩阵的形式$X_n = ((x_{L_X(i, j)}))_{1 \le i, j \le n}$和$Y_n = ((y_{L_Y(i, j)}))_{1 \le i, j \le n}$，在一定的假设下的\emph{link-functions} $L_X$和$L_Y$，最值得注意的是内射的地图$(i, j) \mapsto (L_X(i, j), L_Y(i, j))$。基于数值证据，我们猜想循环律$μ_{\mathrm{circ}}$，即$\mathbb{C}$的单位圆盘上的一致测度，也就是循环变量的Brown测度，实际上是$n^{-1/2}M_n$的极限谱测度.
英文摘要：
Let $\{x_α\}_{α\in \mathbb{Z}}$ and $\{y_α\}_{α\in \mathbb{Z}}$ 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 $(\mathcal{M}_n(\mathbb{C}), \frac{1}{n}\mathrm{tr})$, 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 $\{x_α\}$ and $\{y_β\}$ are independent strongly multiplicative systems (à la Gaposhkin [7]) satisfying an additional \emph{admissibility} condition, and have uniformly bounded moments of all orders -- a nontrivial example of such a system being $\{\sqrt{2}\sin(2^n πU)\}_{n \in \mathbb{Z}_+}$, where $U \sim \mathrm{Uniform}(0, 1)$. 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 \emph{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 $μ_{\mathrm{circ}}$, i.e. 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 of $n^{-1/2}M_n$.
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