In this paper, we analyze the covariance kernel of the Gaussian process that arises as the limit of fluctuations of linear spectral statistics for Wigner matrices with a few moments. More precisely, the process we study here corresponds to Hermitian matrices with independent entries that have $\alpha$ moments for $2<\alpha <4$. We obtain a closed form $\alpha$-dependent expression for the covariance of the limiting process resulting from fluctuations of the Stieltjes transform by explicitly integrating the known double Laplace transform integral formula obtained in [F. Benaych-Georges and A. Maltsev, Fluctuations of linear statistics of half-heavy-tailed random matrices, Stochastic Process. Appl. 126(11) (2016) 3331–3352]. We then express the covariance as an integral kernel acting on bounded continuous test functions. The resulting formulation allows us to offer a heuristic interpretation of the impact the typical large eigenvalues of this matrix ensemble have on the covariance structure.

在本文中，我们分析了高斯过程的协方差核，该协方差核作为维格纳矩阵线性谱统计波动的极限而出现。更准确地说，我们在这里研究的过程对应于具有独立条目的 Hermitian 矩阵$\alpha$片刻$\mathrm{2个}<\alpha <\mathrm{4个}$. 我们得到一个封闭的形式$\alpha$通过显式积分在 [F. Benaych-Georges 和 A. Maltsev，半重尾随机矩阵线性统计的波动，随机过程。申请  126 (11) (2016) 3331–3352]。然后我们将协方差表示为作用于有界连续测试函数的积分核。由此产生的公式使我们能够对该矩阵系综的典型大特征值对协方差结构的影响提供启发式解释。