We give a sufficient condition for the CLT to hold in the space of trace class (nuclear) operators. This condition turns out to be adapted to the asymptotic study of empirical covariance operators of Hilbert valued random variables.

Suppose [mu]n is a sequence of measures on a separable metric space converging weakly to [mu] with rate [pi]n (in the Prohorov metric). We find a new rate of convergence of [mu]nf-1 to [mu]f-1, where f belongs to a wide class of functions between two Banach spaces.