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February 2004 Limit theorems for random normalized distortion
Pierre Cohort
Ann. Appl. Probab. 14(1): 118-143 (February 2004). DOI: 10.1214/aoap/1075828049

Abstract

We present some convergence results about the distortion Dμ,n,rν related to the Voronoï vector quantization of a μ-distributed random variable using n i.i.d. ν-distributed codes. A weak law of large numbers for nr/dDμ,n,rν is derived essentially under a μ-integrability condition on a negative power of a δ-lower Radon--Nikodym derivative of ν. Assuming in addition that the probability measure μ has a bounded ε-potential, we obtain a strong law of large numbers for nr/dDμ,n,rν. In particular, we show that the random distortion and the optimal distortion vanish almost surely at the same rate. In the one-dimensional setting (d=1), we derive a central limit theorem for nrDμ,n,rν. The related limiting variance is explicitly computed.

Citation

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Pierre Cohort. "Limit theorems for random normalized distortion." Ann. Appl. Probab. 14 (1) 118 - 143, February 2004. https://doi.org/10.1214/aoap/1075828049

Information

Published: February 2004
First available in Project Euclid: 3 February 2004

zbMATH: 1041.60022
MathSciNet: MR2023018
Digital Object Identifier: 10.1214/aoap/1075828049

Subjects:
Primary: 60F05 , 60F15 , 60F25
Secondary: 94A29

Keywords: central limit theorem , distortion , Law of Large Numbers , quantization

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.14 • No. 1 • February 2004
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