Open Access
February 2012 Asymptotics of the maximal radius of an Lr-optimal sequence of quantizers
Gilles Pagès, Abass Sagna
Bernoulli 18(1): 360-389 (February 2012). DOI: 10.3150/10-BEJ333


Let P be a probability distribution on ℝd (equipped with an Euclidean norm |⋅|). Let r > 0 and let (αn)n≥1 be an (asymptotically) Lr(P)-optimal sequence of n-quantizers. We investigate the asymptotic behavior of the maximal radius sequence induced by the sequence (αn)n≥1 defined for every n ≥ 1 by ρ(αn) = max{|a|, aαn}. When card(supp(P)) is infinite, the maximal radius sequence goes to sup {|x|, x ∈ supp(P)} as n goes to infinity. We then give the exact rate of convergence for two classes of distributions with unbounded support: distributions with hyper-exponential tails and distributions with polynomial tails. In the one-dimensional setting, a sharp rate and constant are provided for distributions with hyper-exponential tails.


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Gilles Pagès. Abass Sagna. "Asymptotics of the maximal radius of an Lr-optimal sequence of quantizers." Bernoulli 18 (1) 360 - 389, February 2012.


Published: February 2012
First available in Project Euclid: 20 January 2012

zbMATH: 1245.60052
MathSciNet: MR2888710
Digital Object Identifier: 10.3150/10-BEJ333

Keywords: distribution tail , function with regular variation , maximal radius of a quantizer , optimal quantization , Zador theorem

Rights: Copyright © 2012 Bernoulli Society for Mathematical Statistics and Probability

Vol.18 • No. 1 • February 2012
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