# Convergence rate of optimal quantization grids and application to empirical measure

We study the convergence rate of optimal quantization for a probability measure sequence (μ_n)_n∈N^* on R^d which converges in the Wasserstein distance in two aspects: the first one is the convergence rate of optimal grid x^(n)∈(R^d)^K of μ_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 will study the performance of the optimal grid of the empirical measure of a distribution μ with finite second moment but possibly unbounded support. As an application, we show that the mean performance of the empirical measure of the multidimensional normal distribution N(m, Σ) and of distributions with hyper-exponential tails behave like O(K n/√(n)).

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