Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T18:29:35.987Z Has data issue: false hasContentIssue false
  • English
  • Français

On probability properties of measures of random sets and the asymptotic behavior of empirical distribution functions

Published online by Cambridge University Press:  14 July 2016

Gedalia Ailam
Gedalia Ailam*
Affiliation:
Michigan State University
Get access Rights & Permissions [Opens in a new window]

Extract

Probability properties of the measure of the union of random sets have theoretical as well as practical importance (David (1950), Garwood (1947), Hemmer (1959)). In the present paper we derive asymptotic properties of the distributions of these measures and apply the derived properties to the investigation of the asymptotic behavior of empirical distribution functions. Thus, an asymptotic distribution function for the relative lengths of steps in the empirical distribution function is obtained.

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 1 , April 1968 , pp. 196 - 202
Copyright
Copyright © Sheffield: Applied Probability Trust 

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

[1] Ailam, G. (1966) Moment of coverage and coverage spaces. J. Appl. Prob. 3, 550–555.CrossRefGoogle Scholar
[2] David, F. N. (1950) Two combinatorial tests of whether a sample has come from a given population. Biometrica 37, 97–110.Google ScholarPubMed
[3] Garwood, F. (1947) The variance of overlap of geometrical figures with reference to a bombing problem. Biometrika 34, 1–17.CrossRefGoogle ScholarPubMed
[4] Hammer, P. Chr. (1959) A problem of geometrical probabilities. K. norske Vidensk. Selsk. Forh. 32, 117–120.Google Scholar
[5] Loève, M. (1955) Probability Theory. Van Nostrand, New York.Google Scholar
[6] Saks, S. (1937) Theory of the Integral. 2nd Edition. Hafner Publication Co., New York.Google Scholar
[7] Weiss, L. (1955) The stochastic convergence of a function of sample successive differences. Ann. Math. Statist. 26, 532–536.CrossRefGoogle Scholar