Limit theorems for multi-dimensional random quantizers

Abstract

We consider the $r$-th power quantization error arising in the optimal approximation of a $d$-dimensional probability measure $P$ by a discrete measure supported by the realization of $n$ i.i.d. random variables $X_1,...,X_n$. For all $d \geq 1$ and $r \in (0, \infty)$ we establish mean and variance asymptotics as well as central limit theorems for the $r$-th power quantization error. Limiting means and variances are expressed in terms of the densities of $P$ and $X_1$.  Similar convergence results hold for the random point measures arising by placing at each $X_i, 1 \leq i \leq n,$ a mass equal to the local distortion.