Small-time almost-sure behaviour of extremal processes. (26th June 2017)

• Record Type: Journal Article
• Authors: Maller, Ross A.; Schmidli, Peter C.
• Is Part Of: Advances in applied probability. Volume 49:Number 2(2017)
• Page Start: 411
• Page End: 429
• Publication Date: 2017-06-26
• Subjects: Extremal process, -- upper and lower classes, -- relative stability, -- Lévy flight, -- Lévy dust, -- fractal property; Primary 60G51, -- 60G55, -- 60G70, -- Secondary 60F15; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2017.7
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: An rth-order extremal process Δ ( r ) = (Δ ( r ) t ) t ≥0 is a continuous-time analogue of the r th 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.