Stability of quantization dimension and quantization for homogeneous Cantor measures

Abstract

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable.

Optimal Quantization for Uniform Distributions on Cantor-Like Sets

“…[15], Theorem 1.6 (3), respectively [17], Proposition 5.1) abstained from (35). Zhu [24] also characterized the quantization dimension for a large class of singular distributions on higher dimensional Cantor-like sets.…”

“…The special case of one-dimensional dyadic homogeneous Cantor distributions (d = 1, N = 2) was investigated by Kesseböhmer and Zhu [15] and Kreitmeier [17]. Although not explicitely stated, both conditions (SQ) and (MQED) were necessary in these papers as well.…”

“…[19], Example 5.5) that the quantization dimension needs not to exist for singular distributions that are not self-similar. Kesseböhmer and Zhu [15] studied the quantization dimension for homogeneous singular distributions on Cantor-like sets in the 1-dimensional case. To this end, they determined the structure of the optimal sets under the restriction that c = sup i∈N c i ≤ 1 4 .…”

“…Subsequently, for more generalized Cantor distributions, which are not necessarily self-similar, among other results the optimal codebooks were determined by Kesseböhmer and Zhu [15]. Kreitmeier [17] extended this codebook result to a class of Cantor distributions, including the classical one studied by Graf and Luschgy.…”

“…[19], Example 5.5). Later on, the existence of the quantization dimension for distributions on (not necessarily self-similar) Cantor-like sets was systematically studied and characterized by Kesseböhmer and Zhu [15], Kreitmeier [17] and Zhu [24].…”