Tid: 20 mars 2000 kl 1515-1700

Plats : Seminarierummet 3733, Institutionen för matematik, KTH, Lindstedts väg 25, plan 7. Karta!

Föredragshållare: Patrik Albin, Matematisk statistik, Chalmers Tekniska Högskola. Publikationslista.

Titel: On extremes and streams of upcrossings


Let $\,\{\xi(t)\}_{t\in[0,h]}\,$ be a real-valued and ${\bold P}
 $-continuous (continuous in probability) stationary stochastic process. We study and characterize asymptotic (as eg, $\,u\!\to\!\infty$)$\,$ relations between the probability

{\bold P}\bigl\{\sup\nolimits_{t\in[0,h]}\xi(t)\!\gt\!u\bigr\}

of an exceedance of a high level $\,u\,$ and the expression

{\bold P}\{\xi(0)\!\gt\!u\}\,{}^{\!}+\,{}^{\!}h\,{}^{\!}\lim...
 ...\,}{\bold P}\bigl\{\xi(0)\!\le\!u\!<\!\xi(2^{-n})

This latter expression can be interpreted as

Probability $\,\xi(t)\,$ starts above $\,u$ + $\,h{}^{\!}\times{}^{\!}$ upcrossing intensity

so it is natural to expect a relation to the probability of exceedance of the level $\,u$.

Examples of Application include Markov jump processes, stable processes, and quadratic functionals of Gaussian processes.

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