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.