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.

中文翻译：

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关于独立非对称随机矩阵的 Schur-Hadamard 积的 *-收敛性

设 $\{x_{\alpha }\}_{\alpha \in {\mathbb {Z}}}$ 和 $\{y_{\alpha }\}_{\alpha \in {\mathbb {Z}} }$ 是两个独立的零均值、单位方差随机变量集合，具有所有阶的一致有界矩。考虑一个非对称 Toeplitz 矩阵 $X_n = ((x_{i - j}))_{1 \le i, j \le n}$ 和一个汉克尔矩阵 $Y_n = ((y_{i + j}))_{ 1 \le i, j \le n}$，令 $M_n = X_n \odot Y_n$ 为它们的元素/舒尔-哈达玛积。在本文中，我们几乎可以肯定地证明，$n^{-1/2}M_n$，作为 *-概率空间的一个元素 $(\mathcal {M}_n({\mathbb {C}})，\ frac {1}{n}\text {tr})$，在 * 分布中收敛到一个循环变量。使用 iid Rademacher 条目，这种构造给出了一个仅具有 $O(n)$ 位随机性的循环变量的矩阵模型。我们还考虑了一个依赖设置，其中 $\{x_{\alpha }\}$ 和 $\{y_{\beta }\}$ 是满足附加可接纳性条件的独立强乘法系统（à la Gaposhkin [7]），并且具有所有阶的一致有界矩——这种系统的一个重要示例是 $\{\sqrt {2}\sin (2^n \pi U)\}_{n \in {\mathbb {Z}}_+ }$，其中 $U \sim \textrm {统一}(0, 1)$。在这种情况下，我们展示了 $n^{-1/2}M_n$ 的 *-矩与循环变量的非期望收敛和概率收敛。最后，我们将我们的结果推广到形式为 $X_n = ((x_{L_X(i, j)}))_{1 \le i, j \le n}$ 和 $Y_n 的结构化随机矩阵的 Schur-Hadamard 积= ((y_{L_Y(i, j)}))_{1 \le i, j \le n}$，在链接函数 $L_X$ 和 $L_Y$ 的某些假设下，最值得注意的是地图 $(i, j) \mapsto (L_X(i, j), L_Y(i, j))$。根据数值证据，我们推测循环定律$\mu_{\textrm {circ}}$，即${\mathbb {C}}$单位盘上的统一测度，也就是布朗一个循环变量的度量，实际上是 $n^{-1/2}M_n$ 的极限谱度量 (LSM)。如果为真，这将提供一个有趣的示例，其中只有 $O(n)$ 位随机性的随机矩阵将循环定律作为其 LSM。