Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T13:09:16.306Z Has data issue: false hasContentIssue false
  • English
  • Français

On the weak convergence of U-statistic processes, and of the empirical process

Published online by Cambridge University Press:  24 October 2008

R. M. Loynes
R. M. Loynes
Affiliation:
University of Sheffield
Get access Rights & Permissions [Opens in a new window]

1. summary and introduction

In (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 2 , March 1978 , pp. 269 - 272
Copyright
Copyright © Cambridge Philosophical Society 1978

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

REFERENCES

(1)Barbour, A. D.Tail sums of convergent series of independent random variables. Proc. Cambridge Philos. Soc. 75 (1974), 361–364.CrossRefGoogle Scholar
(2)Birnbaum, Z. W. and Marshall, A. W.Some multivariate Chebyshev inequalities with extensions to continuous parameter processes. Ann. Math. Statist. 32 (1961), 687–703.CrossRefGoogle Scholar
(3)Hoeffding, W.A class of statistics with asymptotically normal distributions. Ann. Math. Statist. 19 (1948), 293–325.CrossRefGoogle Scholar
(4)Kiefer, J.Skorokhod embedding of multivariate RV's, and the sample DF. Z. Wahrschein-lichkeitstheorie und Verw. Gebiete 24 (1972), 1–35.CrossRefGoogle Scholar
(5)Loynes, R. M.An invariance principle for reversed martingales. Proc. Amer. Math. Soc. 25 (1970), 56–64.CrossRefGoogle Scholar
(6)Loynes, R. M.A criterion for tightness for a sequence of martingales. Ann. Prob. 4 (1976), 859–862.CrossRefGoogle Scholar
(7)Miller, R. G. Jr. and Sen, P. K.Weak convergence of U-statistics and von Mises' differentiable statistical functions. Ann. Math. Statist. 43 (1972), 31–41.CrossRefGoogle Scholar
(8)Müller, D. W.Verteilungs-Invarianzprinzipien für das starke Gesetz der grossen Zahl. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10 (1968), 173–192.CrossRefGoogle Scholar
(9)Whitt, W.Stochastic Abelian and Tauberian theorems. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 22 (1972), 251–267.CrossRefGoogle Scholar