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 Electronic Communications in Probability > Vol. 13 (2008) > Paper 48 open journal systems 


Limit theorems for multi-dimensional random quantizers

Joseph Yukich, Lehigh University


Abstract
We consider the $rth$ power quantization error arising in the optimal approximation of a $d$-dimensional probability measure $P$ by a discrete measure supported by the realization of $n$ i.i.d. random variables $X_1,...,X_n$. For all $d ≥ 1$ and $r in (0, ∞)$ we establish mean and variance asymptotics as well as central limit theorems for the $rth$ power quantization error. Limiting means and variances are expressed in terms of the densities of $P$ and $X_1$.  Similar convergence results hold for the random point measures arising by placing at each $X_i, 1 ≤ i ≤ n,$ a mass equal to the local distortion.


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Pages: 507-517

Published on: October 13, 2008


Bibliography
  1. Baryshnikov,Y. ; Penrose,M. D. and Yukich, J. E. (2008), Gaussian limits for generalized spacings, arXiv:0804.4123 [math.PR]; condensed version to appear in Ann. Appl. Prob.
  2. Baryshnikov, Yu.; Yukich, J. E. Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15 (2005), no. 1A, 213--253. MR2115042 (2005j:60043)
  3. Billingsley, Patrick. Convergence of probability measures.John Wiley & Sons, Inc., New York-London-Sydney 1968 xii+253 pp. MR0233396 (38 #1718)
  4. Bucklew, James A.; Wise, Gary L. Multidimensional asymptotic quantization theory with $r$th power distortion measures. IEEE Trans. Inform. Theory 28 (1982), no. 2, 239--247. MR0651819 (83b:94026)
  5. Cohort, Pierre. Limit theorems for random normalized distortion. Ann. Appl. Probab. 14 (2004), no. 1, 118--143. MR2023018 (2004j:60073)
  6. Gersho, Allen. Asymptotically optimal block quantization. IEEE Trans. Inform. Theory 25 (1979), no. 4, 373--380. MR0536229 (80h:94023)
  7. Graf, Siegfried; Luschgy, Harald. Foundations of quantization for probability distributions.Lecture Notes in Mathematics, 1730. Springer-Verlag, Berlin, 2000. x+230 pp. ISBN: 3-540-67394-6 MR1764176 (2001m:60043)
  8. Penrose, Mathew D. Gaussian limits for random geometric measures. Electron. J. Probab. 12 (2007), 989--1035 (electronic). MR2336596
  9. Penrose,M. D. (2007), Laws of large numbers in stochastic geometry with statistical applications, Bernoulli, 13, 4, 1124-1150.
  10. Penrose, Mathew D.; Yukich, J. E. Central limit theorems for some graphs in computational geometry. Ann. Appl. Probab. 11 (2001), no. 4, 1005--1041. MR1878288 (2002k:60068)
  11. Penrose, Mathew D.; Yukich, J. E. Weak laws of large numbers in geometric probability. Ann. Appl. Probab. 13 (2003), no. 1, 277--303. MR1952000 (2004b:60034)
  12. Penrose, Mathew D.; Yukich, J. E. Normal approximation in geometric probability. Stein's method and applications, 37--58, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 5, Singapore Univ. Press, Singapore, 2005. MR2201885 (2007f:60015)
  13. Schreiber, T. (2008), Limit theorems in stochastic geometry, New Perspectives in Stochastic Geometry, Oxford University Press, to appear.
  14. Zador, P. L. (1966), Asymptotic quantization error of continuous random variables, unpublished preprint, Bell Laboratories.
  15. Zador, Paul L. Asymptotic quantization error of continuous signals and the quantization dimension. IEEE Trans. Inform. Theory 28 (1982), no. 2, 139--149. MR0651809 (83b:94014)
















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Electronic Communications in Probability. ISSN: 1083-589X