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Quantization of probability distributions under norm-based distortion measures II: Self-similar distributions

TitleQuantization of probability distributions under norm-based distortion measures II: Self-similar distributions
Publication TypeJournal Article
Year of Publication2006
AuthorsSylvain Delattre, Siegfried Graf, Harald Luschgy, and Gilles Pagès
JournalJournal of Mathematical Analysis and Applications
Volume318
Pagination507 - 516
ISSN0022-247X
KeywordsPoint density measure
Abstract

For a probability measure $ P $ on $ \mathbb{R}^d $ and $ n \in \mathbb{N} $ consider $ e_n = \inf \displaystyle \int \min_{a \in \alpha} V(\| x-a \| )dP(x) $ where the infimum is taken over all subsets $ \alpha $ of $ \mathbb{R}^d $ with $ \mbox{card} (\alpha) \leq n $ and $ V $ is a nondecreasing function. Under certain conditions on $ V $, we derive the precise $ n $-asymptotics of $ e_n $ for nonsingular and for (singular) self-similar distributions $ P $ and we find the asymptotic performance of optimal quantizers using weighted empirical measures.