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 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.