Some remarks on absolute continuity and quantization of probability measures
type: Item
instance of: scholarly article
title: Some remarks on absolute continuity and quantization of probability measures
zbMATH Open document ID: 1128.28012
author: Sanguo Zhu
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-05-04T00:00:00Z
review text: Let \(\mathcal M\) be the set of all probability measures on \(({\mathbb R}^d, {\mathcal B})\). A function \(\dim: {\mathcal M} \to {\mathbb R}\) is said to be increasing (respectively decreasing) if for all \(\nu, \mu \in {\mathcal M}\) and \(\nu \ll \mu\) implies \(\dim \nu \leq \dim \mu\) (respectively \(\dim \nu \geq \dim \mu\)). It can be seen that the upper (lower) Hausdorff and packing dimensions are increasing (decreasing). In the paper under review, the author proves that the upper and lower quantization dimensions of order \(r\) are not monotone, and provides sufficient conditions in terms of the so-called vanishing rates such that \(\nu \ll \mu\) implies \(\overline{D}_r(\nu) \leq \overline{D}_r(\mu)\). The author studies the set of quantization dimensions of measures which are absolutely continuous with respect to \(\mu \in {\mathcal M}\), and proves that the infimum on this set coincides with the lower packing dimension \(\dim_{\text{P}} \mu\), of \(\mu\). Moreover, this infimum can be attained provided that \(\dim_{\text{P}}^* \mu = \dim_{\text{P}} \mu.\) As an application, the author also determines the quantization dimension of a class of measures which are absolutely continuous with respect to some self-similar measure \(\mu\).
reviewed by: Y. Xiao
Mathematics Subject Classification ID: 28A80
Mathematics Subject Classification ID: 60E05
zbMATH DE Number: 5149538
zbMATH Keywords: quantization
zbMATH Keywords: quantization dimension
zbMATH Keywords: packing dimension
zbMATH Keywords: absolutely continuity
zbMATH Keywords: monotonicity
zbMATH Keywords: self-similarity
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.08.073
OpenAlex ID: W2018717600
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DOI: 10.1016/J.JMAA.2006.08.073
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description: scientific article; zbMATH DE number 5149538