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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


We present some convergence results about the distortion $\mathcal{D}_{\mu,n,r}^{\nu}$ related to the Voronoï vector quantization of a $\mu$-distributed random variable using $n$ i.i.d. $\nu$-distributed codes. A weak law of large numbers for $n^{r/d}\mathcal{D}_{\mu,n,r}^{\nu}$ is derived essentially under a $\mu$-integrability condition on a negative power of a $\delta$-lower Radon--Nikodym derivative of $\nu$. Assuming in addition that the probability measure $\mu$ has a bounded $\varepsilon$-potential, we obtain a strong law of large numbers for $n^{r/d} \mathcal{D}_{\mu,n,r}^{\nu}$. 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 $n^{r}\mathcal{D}_{\mu,n,r}^{\nu}$. The related limiting variance is explicitly computed.


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Pierre Cohort. "Limit theorems for random normalized distortion." Ann. Appl. Probab. 14 (1) 118 - 143, February 2004.


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

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

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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