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Small-time almost-sure behaviour of extremal processes

  • Ross A. Maller (a1) and Peter C. Schmidli (a1)

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 Δ t (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 lim t b t = 0. We are then able to give an integral criterion for the almost sure relative stability of Δ t (r) as t ↓ 0, r = 1, 2, . . ., or, equivalently, as it turns out, for the almost sure relative stability of Δ t (1) as t ↓ 0.

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Corresponding author

* Postal address: Research School of Finance, Actuarial Studies and Statistics, The Australian National University, Canberra, ACT 0200, Australia.
*** Email address: ross.maller@anu.edu.au
** Email address: peter.schmidli.stats@gmail.com

References

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