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Quantization of probability distributions under norm-based distortion measures. (English) Zbl 1066.60026

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

MSC:

60E99 Distribution theory
94A29 Source coding
28A80 Fractals
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