Small-time almost-sure behaviour of extremal processes

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.

Functional laws for trimmed Lévy processes

“…where Y is an a.s. finite, non-degenerate, non-normal 1 random variable. Then (10) implies that the two-sided tail Π X of X is regularly varying (at 0 or ∞, as appropriate), with index α ∈ (0, 2).…”

“…Theorem 3. Assume that (X t ) t≥0 is in the domain of attraction of a stable law at 0 with nonstochastic centering and norming functions a t ∈ R, b t > 0, so that (9) and (10) hold. In the following, convergences are with respect to the J 1 -topology in D.…”

“…(i) We state proofs just for t ↓ 0; t → ∞ is very similar. Recall the definition of { t (τ )} τ ∈[0,1] from (10) and assume the convergence of t to a stable process Y as in (10). The process Y has Lévy measure Y which is diffuse (continuous at each x ∈ R \ {0}).…”

“…where Y is an a.s. finite, non-degenerate, non-normal 1 random variable. Then (10) implies that 1 When Y is N (0, 1), a standard normal random variable, the large jumps are asymptotically negligible with respect to bt and ( 12) and ( 13) remain true with (r,s) Y and (r) Y a standard Brownian motion; see Fan [4].…”

“…The small time case extends our understanding of local properties of the process and can have direct application as for example in Maller and Fan [8] and Maller and Schmidli [10]; the large time case has the statistical applications alluded to, such as the robustness of statistics, and insurance modelling, etc., as we discuss later.…”