Deterministic Noises that can be Statistically Distinguished from the Random Ones

  • Youri DavydovEmail author
  • Ričardas Zitikis


Empirical measures generated by random sequences with deterministic and random noises have same asymptotic distributions provided that the noises have same asymptotic distributions (cf., Davydov and Zitikis, 2004, Proc. Am. Math. Soc. 132, 1203–1210). This phenomenon has raised an intriguing question about the possibility of distinguishing the two types of noises based only on their asymptotic distributions. In the present paper we suggest an answer to the question by considering asymptotic variances, and distributions, of the appropriately centered and normalized empirical measures and processes.


Empirical measures empirical processes weak convergence asymptotic normality deterministic noise random noise white noise 

AMS Mathematics Subject Classifications 2000

Primary 60F05 60H40 Secondary 62G10 62G20 


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  1. Billingsley, P. 1968Convergence of Probability MeasuresWileyNew YorkzbMATHGoogle Scholar
  2. Bosq, D., Guégan, D. 1995Nonparametric estimation of the chaotic function and the invariant measure of a dynamical systemStatist. Probab. Lett.25201212zbMATHMathSciNetCrossRefGoogle Scholar
  3. Davydov, Y.: Weak convergence of discontinuous processes to continuous ones. In: St. Petersburg (ed), Probability Theory and Mathematical Statistics Gordon and Breach, Amsterdam, 1996, pp. 15–18.Google Scholar
  4. Davydov, Y., Zitikis, R. 2002Convergence of generalized Lorenz curves based on stationary ergodic random sequences with deterministic noiseStatist. Probab. Lett.59329340zbMATHMathSciNetCrossRefGoogle Scholar
  5. Davydov, Y., Zitikis, R. 2004The influence of deterministic noise on empirical measures generated by stationary processesProc. Am. Math. Soc.13212031210zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Laboratoire Paul PainleveUMR 8524, Université des Sciences et Technologies de LilleVilleneuve d’ Ascq CedexFrance
  2. 2.Department of Statistical and Actuarial SciencesUniversity of Western OntarioLondonCanada

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