  We study the convergence rate of the optimal quantization for a probability
measure sequence $(\mu_{n})_{n\in\mathbb{N}^{*}}$ on $\mathbb{R}^{d}$
converging in the Wasserstein distance in two aspects: the first one is the
convergence rate of optimal quantizer $x^{(n)}\in(\mathbb{R}^{d})^{K}$ of
$\mu_{n}$ at level $K$; the other one is the convergence rate of the distortion
function valued at $x^{(n)}$, called the "performance" of $x^{(n)}$. Moreover,
we also study the mean performance of the optimal quantization for the
empirical measure of a distribution $\mu$ with finite second moment but
possibly unbounded support. As an application, we show that the mean
performance for the empirical measure of the multidimensional normal
distribution $\mathcal{N}(m, \Sigma)$ and of distributions with
hyper-exponential tails behave like $\mathcal{O}(\frac{\log n}{\sqrt{n}})$.
This extends the results from [BDL08] obtained for compactly supported
distribution. We also derive an upper bound which is sharper in the
quantization level $K$ but suboptimal in $n$ by applying results in [FG15].
