Abstract
Harel (1980) established the weak convergence of the multivariate truncated empirical processes under φ-mixing conditions for weight functions which vanish on the lower boundary of [0,1]k+1. In this paper we extend the results under strong mixing conditions and also when the weight functions not only vanish on the lower boundary of [0,1]k+1 but also on the upper corner of [0,1]k+1.
Research supported by the Office of Naval Research Contract N0001485-K-0648.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
AHMAD, I. and LIN, P.E. (1980). On the Chernoff-Savage theorem for dependent random sequences. Ann. Inst. Stat. Math. 32, 211–222.
BALACHEFF, S. and DUPONT, G. (1980). Normalité asymptotique des processus empiriques tronqués et des processus de rang. Lecture Notes in Mathematics. Springer Verlag 821. 19–45.
BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York.
DOUKHAN, P. and PORTAL, F. (1983). Moment de variables aléatoires mélangeantes. C.R. Acad. Sc. Paris 297. 129–132.
EINMAHL, J.H.J., RUYMGAART, F.H. and WELLNER, J.A. (1984). Criteria for weak convergence of weighted multivariate empirical processes indexed by points or rectangles. Acta. Sci. Math. (Szeged).
FEARS, T.R. and MEHRA, K. (1974). Weak convergence of a two sample empirical process and a Chernoff-Savage theorem for φ-mixing sequences. Ann Stat. 2, 586–596.
Harel, M. (1980). Convergence en loi pour la topologie de Skorohod du processus empirique multidimensionnel normalisé tronqué et semi-corrigé. Lecture Notes in Mathematics. Springer Verlag, 821, 46–85.
MEHRA, K.L. and RAO, M.S. (1975). Weak convergence in dq-metrics of multidimensional empirical processes. Preprint, University of Alberta, Edmonton, Canada.
NEUHAUS, G. (1971). On weak convergence of stochastic processes with multidimensional time parameter. Ann Statist. 42, 1285–1295.
NEUMANN, N. (1982). Ein Schwaches Invarianzprinzip für den gewichteten empirischen Prozess von gleichmässig mischenden Zufallsvariablen. Ph.D. Thesis, Göttingen, W. Germany.
WITHERS, C.S. (1975). Convergence of empirical processes of mixing rv’s on [0,1]. Ann. Stat. 3, 1101–1108.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1987 D. Reidel Publishing Company
About this chapter
Cite this chapter
Harel, M., Puri, M.L. (1987). Weak Convergence of Weighted Multivariate Empirical Processes Under Mixing Conditions. In: Puri, M.L., Révész, P., Wertz, W. (eds) Mathematical Statistics and Probability Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3963-9_11
Download citation
DOI: https://doi.org/10.1007/978-94-009-3963-9_11
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-8258-7
Online ISBN: 978-94-009-3963-9
eBook Packages: Springer Book Archive