On the asymptotics of quantizers in two dimensions (Q1364668)

Optimal quantizers of the random vector \(X\) distributed over a region \(D \subset \mathbb{R}^d\) are a finite set of points in \(D\) such that the \(\gamma\)th mean distance of the random vector from this set is minimized. For \(\gamma =2\) and uniform bivariate random vectors, asymptotically optimal quantizers correspond to the centers of regular hexagons [\textit{D. J. Newman}, IEEE Trans. Inf. Theory, IT-28, 137-139 (1982; Zbl 0476.94006)]. The distance to minimize is \(E\| X-T_N \|^\gamma\), where \(\gamma >0\), norm is Euclidean, expectation is with respect to a density function \(p\), \(T_N\) is a set of \(N\) points \(x_{iN}\), and expectations are computed over the Voronoi regions \(D_{iN}= \{x\in D\) and \(\| x-x_{iN} \|= \min_{1\leq j \leq N} \| x-x_{jN}\|\}\). This paper considers bivariate random vectors with finite \(\gamma\)th moments, and a complete characterization of the asymptotically optimal quantizers is given. It is also shown that a related procedure is asymptotically optimal for every \(\gamma>0\). Examples with normal and Pearson type VII distributions are considered.