AI 中文总结
该研究证明了由有限四阶矩独立同分布元素构造的归一化随机矩阵元组，几乎必然强收敛到匹配方差的自由圆族*-分布，通过相关普适性估计、Anderson定理及Bai-Yin范数界完成推导。
AI 中文摘要
设\boldsymbol{X}_{1,n},\boldsymbol{X}_{d,n}是由独立同分布的元素阵列构造的n×n随机矩阵，其元素为中心化的，且按n^{-1/2}归一化。我们证明：若每个元素分布律都具有有限四阶矩，则该矩阵元组几乎必然在*-分布意义下强收敛到具有匹配方差的自由圆族。等价地，对于每个固定的非交换*-多项式（包括带有固定矩阵系数的多项式），归一化的迹和算子范数均收敛。对于复元素的伪方差，我们未施加任何假设。有界元素论证将Brailovskaya和van Handel的谱与矩普适性估计应用于所有自伴线性 pencil（线性束）。匹配的高斯 pencil 可约化为独立Wigner矩阵，并通过Anderson强收敛定理予以识别。固定水平的中心化截断结合Bai-Yin范数界，将该结果推广到有限四阶矩的情形。
英文摘要
Let \(X_{1,n},\ldots,X_{d,n}\) be \(n\times n\) random matrices built from independent i.i.d. entry arrays, with centered entries, normalized by \(n^{-1/2}\). We prove that, if every entry law has finite fourth moment, then this tuple converges almost surely strongly in \(*\)-distribution to a free circular family with the matching variances. Equivalently, normalized traces and operator norms converge for every fixed noncommutative \(*\)-polynomial, including polynomials with fixed matrix coefficients. No assumption is imposed on the pseudo-variances of the complex entries. The bounded-entry argument applies the spectrum and moment universality estimates of Brailovskaya and van Handel to all self-adjoint linear pencils. The matching Gaussian pencils are reduced to independent Wigner matrices and identified by Anderson's strong convergence theorem. A fixed-level centered truncation, followed by the Bai--Yin norm bound, transfers the result to finite fourth moments.
Comments21 pages
