Limit theorems for random normalized distortion

We present some convergence results about the distortion D..,n,r related to the Voronoï vector quantization of a .-distributed random variable using n i.i.d. .-distributed codes. A weak law of large numbers for nr/dD..,n,r is derived essentially under a .-integrability condition on a negative power of a .-lower Radon--Nikodym derivative of .. Assuming in addition that the probability measure . has a bounded .-potential, we obtain a strong law of large numbers for nr/dD..,n,r. 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 nrD..,n,r. The related limiting variance is explicitly computed.