The Annals of Applied Probability

Limit theorems for random normalized distortion

Pierre Cohort

Full-text: Open access

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

Subjects
Primary: 60F25: $L^p$-limit theorems 60F15: Strong theorems 60F05: Central limit and other weak theorems
Secondary: 94A29: Source coding [See also 68P30]

Keywords
Quantization distortion law of large numbers central limit theorem

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


Export citation

References

  • Benveniste, A., Métivier, M. and Priouret, P. (1990). Adaptative Algorithms and Stochastic Approximations. Springer, Berlin.
  • Bouton, C. and Pagès, G. (1994). Convergence in distribution of the one-dimensional Kohonen algorithms when the stimuli are not uniform. Adv. in Appl. Probab. 26 80--103.
  • Bouton, C. and Pagès, G. (1997). About the multidimensional competitive learning vector quantization algorithm with constant gain. Ann. Appl. Probab. 7 679--710.
  • Bucklew, J. A. (1981). Companding and random quantization in several dimensions. IEEE Trans. Inform. Theory 27 207--211.
  • Bucklew, J. A. and Wise, G. (1982). Multidimensional asymptotic quantization theory with $r$th power distortion measures. IEEE Trans. Inform. Theory 28 239--247.
  • Conway, J. H. and Sloane, J. A. (1982). Voronoï regions of lattices, second moments of polytopes and quantization. IEEE Trans. Inform. Theory 28 211--226.
  • Fort, J. C. and Pagès, G. (1995). About the a.s. convergence of the Kohonen algorithm with a general neighborhood function. Ann. Appl. Probab. 5 1177--1216.
  • Gersho, A. (1979). Asymptotically optimal block quantization. IEEE Trans. Inform. Theory 25 373--380.
  • Gersho, A. and Gray, R. M. (1992). Vector Quantization and Signal Compression, 7th ed. Kluwer, Dordrecht.
  • Graf, S. and Luschgy, H. (1999). Foundations of Quantization for Probability Distributions. Springer, Berlin.
  • Gray, R. M. and Neuhoff, D. L. (1998). Quantization. IEEE Trans. Inform. Theory 44 2325--2383.
  • Hall, P. (1984). Limit theorems for sums of general functions of $m$-spacing. Math. Proc. Cambridge Philos. Soc. 96 517--532.
  • Hall, P. and Heyde, C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York.
  • Lloyd, S. P. (1982). Least squares quantization in PCM. IEEE Trans. Inform. Theory 28 129--137.
  • Na, S. and Neuhoff, D. L. (1995). Bennett's integral for vector quantizer. IEEE Trans. Inform. Theory 41 886--899.
  • Pagès, G. (1997). A space vector quantization for numerical integration. J. Appl. Comput. Math. 89 1--38.
  • Pagès, G., Bally, V. and Printems, J. (2001). A stochastic quantization method for nonlinear problems. Monte Carlo Methods Appl. 7 21--34.
  • Pyke, R. (1965). Spacings. J. Roy. Statist. Soc. Ser. B 27 395--449.
  • Sabin, M. J. and Gray, R. M. (1986). Global convergence and empirical consistency of the generalized Lloyd algorithm. IEEE Trans. Inform. Theory 32 148--155.
  • Trushkin, A. V. (1993). On the design of an optimal quantizer. IEEE Trans. Inform. Theory 39 1180--1194.
  • Zador, P. L. (1982). Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28 139--149.