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

Convergence rate of optimal quantization grids and application to empirical measure