Precise asymptotics for complete moment convergence in Hilbert spaces

Let {X, X-n; n >= 1} be a sequence of i.i.d, random variables taking values in a real separable Hilbert space (H, parallel to.parallel to) with covariance operator Sigma. Set S-n = Sigma(n)(i=1) X-i, n >= 1. We prove that for 1 < p < 2 and r > 1 + p/2, lim(epsilon SE arrow 0)epsilon((2r-p-2)/(2-p)) Sigma(infinity)(n=1)n(r/p-2-1/p)E{parallel to S-n parallel to-sigma epsilon n(1/p)}+ =sigma(-(2r-2-p)/(2-p))p(2-p)/(r-p)(2r-p-2)E parallel to gamma parallel to(2(r-p)/(2-p)), where gamma is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator Sigma, and sigma(2) is the largest eigenvalue of Sigma.