The Annals of Probability

Rates of convergence for the empirical quantization error

Siegfried Graf and Harald Luschgy

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Abstract

For $n, k \in \mathbb{N}$ and $r > 0$ let $e_{n,r}(P_k)^r = \inf 1/k \sum_{i=1}^k ||X_i - f(X_i)||^r$, where the infimum is taken over all measurable maps $f : \mathbb{R}^d \to \mathbb{R}^d$ with $|f(\mathbb{R}^d)| \leq n$ and $X_1, \dots, X_k$ are i.i.d. $\mathbb{R}^d$-valued random variables. We analyse the asymptotic a.s. behaviour of the $n$th empirical quantization error $e_{n,r}(P_k)$.

Article information

Source
Ann. Probab., Volume 30, Number 2 (2002), 874-897.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481010

Digital Object Identifier
doi:10.1214/aop/1023481010

Mathematical Reviews number (MathSciNet)
MR1905859

Zentralblatt MATH identifier
1018.60032

Subjects
Primary: 60F15: Strong theorems 60E15: Inequalities; stochastic orderings
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20] 94A29: Source coding [See also 68P30]

Keywords
multidimensional quantization $L_r$-error empirical measure empirical quantization error empirical process

Citation

Graf, Siegfried; Luschgy, Harald. Rates of convergence for the empirical quantization error. Ann. Probab. 30 (2002), no. 2, 874--897. doi:10.1214/aop/1023481010. https://projecteuclid.org/euclid.aop/1023481010


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