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We present a probabilistic analysis of two Krylov subspace methods forsolving linear systems. We prove a central limit theorem for norms of theresidual vectors that are produced by the conjugate gradient and MINRESalgorithms when applied to a wide class of sample covariance matricessatisfying some standard moment conditions. The proof involves establishing afour moment theorem for the so-called spectral measure, implying, inparticular, universality for the matrix produced by the Lanczos iteration. Thecentral limit theorem then implies an almost-deterministic iteration count forthe iterative methods in question.

Universality for the conjugate gradient and MINRES algorithms on sample covariance matrices