论文标题
关于$*$ - 独立非对称随机矩阵的Schur-Hadamard产品的收敛
On $*$-Convergence of Schur-Hadamard Products of Independent Nonsymmetric Random Matrices
论文作者
论文摘要
令$ \ {x_α\} _ {α\ in \ mathbb {z}} $和$ \ {y_α\} _ {α{α\ in \ mathbb {z}} $是两个独立的零均值，单位变异随机变量的独立集合，具有均匀限制的所有级数。考虑一个非对称toeplitz矩阵$ x_n =（（（x_ {i -j}））_ {1 \ le i，j \ le n} $和hankel matrix $ y_n =（（y__ {y_ {i + j}）） ElementWise/Schur-Hadamard产品。在本文中，我们几乎可以肯定地表明，$ n^{ - 1/2} m_n $，是$*$ - 概率空间$（\ Mathcal {m} _n（\ Mathbb {C}），\ frac {1} {1} {n} {n} {n} {n} \ mathrm {tr}）$，$*$*$*$*。与I.I.D. Rademacher条目，该结构为只有$ o（n）$随机性的圆形变量提供了矩阵模型。我们还考虑了一个因设置，其中$ \ {x_α\} $和$ \ {y_β\} $是独立的强度乘法系统（àlagaposhkin [7]），可满足额外的\ emph {可接受性}条件，并且具有均匀界限的所有顺序 - $ \ {\ sqrt {2} \ sin（2^nπu）\} _ {n \ in \ mathbb {z} _+} $，其中$ \ sim \ sim \ sim \ sim \ mathrm {cormiform {roformaniform}（0，1）$。在这种情况下，我们向圆形变量的$ n^{ - 1/2} m_n $的$*$ - 矩矩阵的不可估量和概率收敛。最后，我们将结果概括为$ x_n =（（x__ {l_x（i，j）}）的结构性随机矩阵的Schur-Hadamard产品）_ {1 {1 \ le i，j \ le n} $ and $ y_n =（（y__y（y y y y n y y y n y y ly n y le n y y le n y le n ot） \ emph {link-fintsions} $ l_x $和$ l_y $，最著名的是地图$（i，j）\ mapsto（l_x（i，j），l_y（i，j））$的注射率。根据数值证据，我们猜测，循环定律$μ_ {\ mathrm {circ}} $，即$ \ mathbb {c} $的单位磁盘上的统一度量，这也是循环变量的棕色度量，实际上是$ n^{ - 1/2} m_ $ n^的极限频谱。
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 $n^{-1/2}M_n$.
