Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-24T03:08:46.611Z Has data issue: false hasContentIssue false
  • English
  • Français

A central limit theorem for exchangeable variates with geometric applications

Published online by Cambridge University Press:  14 July 2016

P. A. P. Moran
P. A. P. Moran*
Affiliation:
The Australian National University
Get access Rights & Permissions [Opens in a new window]

Abstract

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

Keywords

CENTRAL LIMIT THEOREMEXCHANGEABLE VARIABLESGEOMETRIC PROBABILITYOCCUPANCY
Type
Research Papers
Information
Journal of Applied Probability , Volume 10 , Issue 4 , December 1973 , pp. 837 - 846
Copyright
Copyright © Applied Probability Trust 1973 

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.)

References

Bernstein, S. (1926–7) Sur l'extension du théorème limité du calcul des probabilités aux sommes des quantités dépendents. Math. Ann. 97, 1–59.CrossRefGoogle Scholar
Bronowski, J. and Neyman, J. (1945) The variance of the measure of a two-dimensional random set. Ann. Math. Statist. 16, 330–341.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York.Google Scholar
Garwood, F. (1947) The variance of the overlap of geometrical figures with reference to a bombing problem. Biometrika, 34, 1–17.CrossRefGoogle ScholarPubMed
Holst, L. (1972) Asymptotic normality and efficiency for certain goodness-of-fit tests. Biometrika 59, 137–145.CrossRefGoogle Scholar
Melnyk, T. W. and Rowlinson, J. S. (1971) The statistics of the volumes covered by systems of penetrating spheres. J. Computational Phys. 7, 385–393.Google Scholar
Moran, P. A. P. (1973) The random volume of interpenetrating spheres in space. J. Appl. Prob. 10, 483–490.Google Scholar
Rényi, A. (1962) Three new proofs and a generalisation of a theorem of Irving Weiss. Publ. Math. Inst. Hung. Acad. Sci. 7, 203–214.Google Scholar
Robbins, H. (1945) On the measure of a random set. II. Ann. Math. Statist. 16, 342–347.Google Scholar
Santaló, L. (1947) On the first two moments of the measure of a random set. Ann. Math. Statist. 18, 37–49.Google Scholar
Votaw, D. F. (1946) The probability distribution of the measure of a random linear set. Ann. Math. Statist. 17, 240–244.Google Scholar
Weiss, I. (1958) Limiting distributions in some occupancy problems. Ann. Math. Statist. 29, 878–884.Google Scholar