Quantization for probability measures

Abstract

We introduce the quantization number and the essential covering rate. We treat the quantization for product measures and give effective upper bounds for the quantization dimension of measures. Complete moment condition and limit quantization dimension are introduced and studied.We introduce and study stability and stabilization for dimensions of measures and prove that the stabilized upper quantization dimension coincides with the packing dimension. The quantization for homogeneous Cantor measures are studied in detail to construct examples showing that the lower quantization dimension is not finitely stable.We introduce the upper and lower vanishing rates and study the relationship between the quantization and absolute continuity of measures. We give conditions to ensure monotonicity of the quantization dimension. Measures which are absolutely continuous with respect to self-similar measures are particularly studied.