Hostname: page-component-89b8bd64d-nlwjb Total loading time: 0 Render date: 2026-05-12T12:03:03.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Limit theorems for dissociated random variables

Published online by Cambridge University Press:  01 July 2016

B. W. Silverman
B. W. Silverman*
Affiliation:
University of Cambridge
Get access Rights & Permissions [Opens in a new window]

Abstract

Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj , …, Yz ) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.

Keywords

CENTRAL LIMIT THEOREMDISTANCE DISTRIBUTIONSIMILARITY MEASURETEST OF RANDOMNESSTEST OF CLUSTERINGCLUSTER ANALYSISDEPENDENT RANDOM VARIABLESWEAK CONVERGENCEEMPIRICAL DISTRIBUTION FUNCTIONGAUSSIAN PROCESSGRAPH COLOURING

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 8 , Issue 4 , December 1976 , pp. 806 - 819
Copyright
Copyright © Applied Probability Trust 1976 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Bartlett, M. S. (1964) The spectral analysis of two-dimensional point processes. Biometrika 51, 299–311.Google Scholar
Billingsley, P. (1968) Convergence of Probability Measures. Wiley, New York.Google Scholar
Cramér, H. and Wold, H. (1936) Some theorems on distribution functions. J. London Math. Soc. 11, 290–295.CrossRefGoogle Scholar
Garsia, A. M. (1970) Continuity properties of Gaussian processes with multidimensional time parameter. Proc. 6th. Berkeley Symp. Math. Statist. Prob. 2, 369–374.Google Scholar
Harary, F. (1969) Graph Theory, Addison-Wesley, Reading, Mass. Google Scholar
Komlos, J., Major, P. and Tusnady, G. (1975) An approximation of partial sums of independent random variables, and the sample distribution function. Z. Wahrscheinlichkeitsth. 32, 111–131.Google Scholar
Loeve, M. (1960) Probability Theory, 2nd edn. Van Nostrand, Princeton.Google Scholar
McGinley, W. G. and Sibson, R. (1975) Dissociated random variables. Math. Proc. Camb. Phil. Soc. 77, 185–188.Google Scholar
Ripley, B. D. (1976) The second order analysis of stationary point processes. J. Appl. Prob. 13, 255–266.CrossRefGoogle Scholar
Silverman, B. W. (1976) Weak and strong uniform consistency of the kernel estimate of a density and its derivatives. Submitted to Ann. Statist. Google Scholar