Small ball probabilities around random centers of Gaussian measures and applications to quantization

Let mu be a centered Gaussian measure on a separable Hilbert space (E, || . ||). We are concerned with the logarithmic small ball probabilities around a mu-distributed center X. It turns out that the asymptotic behavior of -log mu(B(X, epsilon)) is a.s. equivalent to that of a deterministic function phi(R)(epsilon). These new insights will be used to derive the precise asymptotics of a random quantization problem which was introduced in a former article by Dereich, Fehringer, Matoussi, and Scheutzow((8)).