Limit theorems for random normalized distortion

We present some convergence results about the distortion D-mu,n,r(v) related to the Voronoi vector quantization of a mu-distributed random variable using n i.i.d. v-distributed codes. A weak law of large numbers for n(r/d)D(mu,n,r)(v) is derived essentially under a mu-integrability condition on a negative power of a delta-lower Radon-Nikodym derivative of v. Assuming in addition that the probability measure mu has a bounded epsilon-potential, we obtain a strong law of large numbers for n(r/d)D(mu,n,r)(v). In particular, we show that the random distortion and the optimal distortion vanish almost surely at the same rate. In the one-dimensional setting (d = 1), we derive a central limit theorem for n(r)D(mu,n,r)(v). The related limiting variance is explicitly computed.