We study the fluctuations of linear statistics with polynomial test functions for Multiple Orthogonal Polynomial Ensembles. Multiple Orthogonal Polynomial Ensembles form an important class of determinantal point processes that include random matrix models such as the GUE with external source, complex Wishart matrices, multi-matrix models and others.

Our analysis is based on the recurrence matrix for the multiple orthogonal polynomials, that is constructed out of the nearest neighbor recurrences. If the coefficients for the nearest neighbor recurrences have limits, then we show that the right-limit of this recurrence matrix is a matrix that can be viewed as representation of a Toeplitz operator with respect to a non-standard basis. This will allow us to prove Central Limit Theorems for linear statistics of Multiple Orthogonal Polynomial Ensembles. A particular novelty is the use of the Baker–Campbell–Hausdorff formula to prove that the higher cumulants of the linear statistics converge to zero.

We illustrate the main results by discussing Central Limit Theorems for the Gaussian Unitary Ensembles with external source, complex Wishart matrices and specializations of the Schur measure related to multiple Charlier, multiple Krawtchouk and multiple Meixner polynomials.

我们用多项正交检验函数的多项式检验函数研究线性统计量的波动。多个正交多项式集合构成了一类重要的行列式点过程，包括随机矩阵模型，例如带有外部源的GUE，复杂的Wishart矩阵，多矩阵模型等。

我们的分析基于多个正交多项式的递归矩阵，该矩阵是根据最近的邻居递归构造而成的。如果最近邻递归的系数有限制，则表明该递归矩阵的右限是一个矩阵，可以将其视为非标准基础上Toeplitz算符的表示形式。这将使我们能够证明多个正交多项式集合的线性统计的中心极限定理。一个特别的新颖之处是使用Baker-Campbell-Hausdorff公式来证明线性统计量的较高累积量收敛为零。

我们通过讨论具有外部源，复杂Wishart矩阵和与多个Charlier，多个Krawtchouk和多个Meixner多项式相关的Schur度量的特殊化的高斯Unit集的中心极限定理，来说明主要结果。