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