Small-time almost-sure behaviour of extremal processes

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.