## The Annals of Applied Probability

### Limit theorems for random normalized distortion

Pierre Cohort

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab., Volume 14, Number 1 (2004), 118-143.

Dates
First available in Project Euclid: 3 February 2004

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1075828049

Digital Object Identifier
doi:10.1214/aoap/1075828049

Mathematical Reviews number (MathSciNet)
MR2023018

Zentralblatt MATH identifier
1041.60022

#### Citation

Cohort, Pierre. Limit theorems for random normalized distortion. Ann. Appl. Probab. 14 (2004), no. 1, 118--143. doi:10.1214/aoap/1075828049. https://projecteuclid.org/euclid.aoap/1075828049

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