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Small-time almost-sure behaviour of extremal processes

Part of: Stochastic processes Limit theorems

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.
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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 

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