We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four moment theorem for the so-called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost-deterministic iteration count for the iterative methods in question.

中文翻译：

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共轭梯度和 MINRES 算法在样本协方差矩阵上的通用性

我们提出了用于求解线性系统的两种 Krylov 子空间方法的概率分析。我们证明了当应用于满足某些标准矩条件的各种样本协方差矩阵时，由共轭梯度和 MINRES 算法产生的残差向量范数的中心极限定理。证明涉及为所谓的谱测度建立四矩定理，特别是暗示由 Lanczos 迭代产生的矩阵的普遍性。然后，中心极限定理暗示了所讨论的迭代方法的几乎确定性的迭代次数。