Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T13:08:46.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic distribution for the sum and maximum of Gaussian processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  14 July 2016

W. P. McCormick  and
Y. Qi
W. P. McCormick*
Affiliation:
University of Georgia
Y. Qi*
Affiliation:
University of Georgia
*
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA.
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

Previous work on the joint asymptotic distribution of the sum and maxima of Gaussian processes is extended here. In particular, it is shown that for a stationary sequence of standard normal random variables with correlation function r, the condition r(n)ln n = o(1) as n → ∞ suffices to establish the asymptotic independence of the sum and maximum.

Keywords

Gaussian processmaximumsumweakly dependentstrongly dependent

MSC classification

Secondary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 958 - 971
Copyright
Copyright © by the Applied Probability Trust 2000 

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

Chow, T. L., and Teugels, J. L. (1978). The sum and the maximum of i.i.d. random variables. In Proceedings of the Second Prague Symposium on Asymptotic Statistics, eds Mandl, P. and Huškova, M. North Holland, Amsterdam, pp. 81–92.Google Scholar
Ho, H. C., and Hsing, T. (1996). On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. J. Appl. Prob. 33, 138–145.CrossRefGoogle Scholar
Ho, H. C., and McCormick, W. P. (1999). Asymptotic distribution of sum and maximum for strongly dependent Gaussian processes. J. Appl. Prob. 36, 1031–1044.CrossRefGoogle Scholar
Hsing, T. (1995). A note on the asymptotic independence of the sum and maximum of strongly mixing stationary random variables. Ann. Prob. 23, 938–947.CrossRefGoogle Scholar
Leadbetter, M. R., Lindgren, G., and Rootzen, H. (1983). Extremes and Related Properties of Stationary Sequences and Processes. Springer, New York.CrossRefGoogle Scholar
McCormick, W. P. (1980). Weak convergence for the maxima of stationary Gaussian processes using random normalization. Ann. Prob. 8, 483–497.CrossRefGoogle Scholar
McCormick, W. P., and Mittal, Y. (1976). On weak convergence of the maximum. Tech. Rept 81, Dept of Statistics, Stanford University.Google Scholar