  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.
