Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-27T09:30:36.260Z Has data issue: false hasContentIssue false
  • English
  • Français

Ergodic behaviour of extreme values

Part of: Limit theorems

Published online by Cambridge University Press:  09 April 2009

S. Cheng ,
L. Peng  and
Y. Qi
S. Cheng
Affiliation:
Department of Probability and Statistics Peking UniversityBeijing, 100871 P. R. China e-mail: shcheng@pku.edu.cn
L. Peng
Affiliation:
University of Georgia Department of Statistics 220 Statistics Building Athens, Georgia USA e-mail: yai@stat.uga.edu
Y. Qi
Affiliation:
Center for Mathematics and its Applications Australian National UniversityCanberra, ACT 0200 Australia e-mail: liang. peng@maths.anu.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let {Xn, n ≥ 1} be independent identically distributed random variables with a common non-degenerate distribution function F. For each n ≥ 1, denote Mn = max {X1,…, Xn}. Under certain conditions on F, there exist constants an > 0 and bn ∈ R such that . In this paper, we shall show that {(Mn – bn)/an} exhibits ergodic behaviour under additional conditions of F.

Keywords

ergodic behaviourextreme valueslogarithmic meansalmost sure convergence.

MSC classification

Secondary: 60F05: Central limit and other weak theorems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 68 , Issue 2 , April 2000 , pp. 170 - 180
Copyright
Copyright © Australian Mathematical Society 2000

References

[1]Brosamler, G. A., ‘An almost everywhere central limit theorem’, Math. Proc. Camb. Philos. Soc. 104 (1988), 561–574.CrossRefGoogle Scholar
[2]Cheng, S., Peng, L. and Qi, Y., ‘Almost sure convergence in extreme value theory’, Math. Nachr. 190 (1998), 43–50.CrossRefGoogle Scholar
[3]Chow, Y. S. and Teicher, H., Probability theory (Springer, New York, 1988).CrossRefGoogle Scholar
[4]Geluk, J. and de Haan, L., Regular variation, extensions and Tauberian theorems (CWI Tract 40, Amsterdam, 1987).Google Scholar
[5]de Haan, L., On regular variation and its application to the weak convergence of sample extremes (CWI Tract 32, Amsterdam, 1970).Google Scholar
[6]Lacey, M. T. and Philipp, W., ‘A note on the almost sure central limit theorem’, Statist. Probab. Lett. 9 (1990), 201–205.CrossRefGoogle Scholar
[7]Resnick, S. I., Extreme values, regular variation, and point processes (Springer, New York, 1987).CrossRefGoogle Scholar
[8]Schatte, P., ‘On strong versions of the central limit theorem’, Math. Nachr. 137 (1988), 249–256.CrossRefGoogle Scholar
[9]Schatte, P., ‘On the almost sure convergence of subsequences in the central limit theorem’, Statistics 20 (1989), 593–605.CrossRefGoogle Scholar
[10]Schatte, P., ‘Two remarks on the almost sure central limit theorem’, Math. Nachr. 154 (1991), 225–229.CrossRefGoogle Scholar
[11]Schatte, P., ‘On the central limit theorem with almost sure convergence’, Probab. Math. Statist. 11 (1991), 237–246.Google Scholar