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Small-time almost-sure behaviour of extremal processes

Part of:Stochastic processesLimit 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

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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 b t > 0 with limt↓ b t = 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 processes60G55: Point processes60G70: Extreme value theory; extremal processes
Secondary:60F15: Strong theorems

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 2 , June 2017, pp. 411 - 429
Copyright
Copyright © Applied Probability Trust 2017 

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