Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T21:27:06.875Z Has data issue: false hasContentIssue false
  • English
  • Français

Small-time almost-sure behaviour of extremal processes

Part of: Limit theorems Stochastic processes

Published online by Cambridge University Press:  26 June 2017

Ross A. Maller  and
Peter C. Schmidli
Ross A. Maller*
Affiliation:
The Australian National University
Peter C. Schmidli*
Affiliation:
The Australian National University
*
* Postal address: Research School of Finance, Actuarial Studies and Statistics, The Australian National University, Canberra, ACT 0200, Australia.
* Postal address: Research School of Finance, Actuarial Studies and Statistics, The Australian National University, Canberra, ACT 0200, Australia.
Get access Rights & Permissions [Opens in a new window]

Abstract

An rth-order extremal process Δ(r) = (Δ(r)t)t≥0 is a continuous-time analogue of the rth partial maximum sequence of a sequence of independent and identically distributed random variables. Studying maxima in continuous time gives rise to the notion of limiting properties of Δt(r) as t ↓ 0. Here we describe aspects of the small-time behaviour of Δ(r) by characterising its upper and lower classes relative to a nonstochastic nondecreasing function bt > 0 with limt↓bt = 0. We are then able to give an integral criterion for the almost sure relative stability of Δt(r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δt(1) as t ↓ 0.

Keywords

Extremal processupper and lower classesrelative stabilityLévy flightLévy dustfractal property

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G55: Point processes 60G70: Extreme value theory; extremal processes
Secondary: 60F15: Strong theorems
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017 , pp. 411 - 429
Copyright
Copyright © Applied Probability Trust 2017 

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] Barndorff-Nielsen, O. (1961). On the rate of growth of the partial maxima of a sequence of independent identically distributed random variables. Math. Scand. 9, 383–394. Google Scholar
[2] Barndorff-Nielsen, O. (1963). On the limit behaviour of extreme order statistics. Ann. Math. Statist. 34, 992–1002. CrossRefGoogle Scholar
[3] Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopedia Math. Appl. 27). Cambridge University Press. Google Scholar
[4] Brémaud, P. (1981). Point Processes and Queues: Martingale Dynamics. Springer, New York. CrossRefGoogle Scholar
[5] Buchmann, B., Fan, Y. and Maller, R. A. (2016). Distributional representations and dominance of a Lévy process over its maximal jump processes. Bernoulli 22, 2325–2371. Google Scholar
[6] De Haan, L. and Ferreira, A. (2006). Extreme Value Theory: An Introduction. Springer, New York. Google Scholar
[7] Gnedenko, B. (1943). Sur la distribution limite du terme maximum d'une séries aléatoire. Ann. Math. 44, 423–453. CrossRefGoogle Scholar
[8] Godbole, A. P. (1987). On Klass' series criterion for the minimal growth rate of partial maxima. Statist. Prob. Lett. 5, 235–238. CrossRefGoogle Scholar
[9] Goldie, C. M. and Maller, R. A. (1996). A point-process approach to almost-sure behaviour of record values and order statistics. Adv. Appl. Prob. 28, 426–462. CrossRefGoogle Scholar
[10] Hall, P. (1979). On the relative stability of large order statistics. Math. Proc. Camb. Phil. Soc. 86, 467–475. Google Scholar
[11] Klass, M. J. (1984). The minimal growth rate of partial maxima. Ann. Prob. 12, 380–389. Google Scholar
[12] Klass, K. J. (1985). The Robbins-Siegmund series criterion for partial maxima. Ann. Prob. 3, 1369–1370. Google Scholar
[13] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes and Applications. Springer, Heidelberg. Google Scholar
[14] Maller, R. A. (2015). Strong laws at zero for trimmed Lévy processes. Electron. J. Prob. 20, 24pp. CrossRefGoogle Scholar
[15] Resnick, S. I. (1974). Inverses of extremal processes. Adv. Appl. Prob. 6, 392–406. Google Scholar
[16] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York. Google Scholar
[17] Resnick, S. I. and Rubinovitch, M. (1973). The structure of extremal processes. Adv. Appl. Prob. 5, 287–307. Google Scholar
[18] Resnick, S. I. and Tomkins, R. J. (1973). Almost sure stability of maxima. J. Appl. Prob. 10, 387–401. CrossRefGoogle Scholar
[19] Spitzer, F. (1975). Principles of Random Walk. Springer, New York. Google Scholar
[20] Tomkins, R. J. (1986). Regular variation and the stability of maxima. Ann. Prob. 4, 984–995. Google Scholar