Limit theorems for multi-dimensional random quantizers. (English) Zbl 1189.60054

Yukich, Joseph

Electron. Commun. Probab. 13, 507-517 (2008).

Summary: We consider the \(rth\) 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 \(X_1,\dots,X_n\). For all \(d \geq 1\) and \(r \in (0, \infty )\) we establish mean and variance asymptotics as well as central limit theorems for the \(rth\) 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.

MSC:

60F05

Central limit and other weak theorems

Keywords:

quantization; laws of large numbers; central limit theorems; stabilization