On Functional Records and Champions

Clément Dombry (), Michael Falk () and Maximilian Zott ()
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Clément Dombry: Université de Franche-Comté
Michael Falk: Institute of Mathematics
Maximilian Zott: Institute of Mathematics

Journal of Theoretical Probability, 2019, vol. 32, issue 3, 1252-1277

Abstract: Abstract A record among a sequence of iid random variables $$X_1,X_2,\dots $$ X 1 , X 2 , ⋯ on the real line is defined as a member $$X_n$$ X n such that $$X_n>\max (X_1,\cdots ,X_{n-1})$$ X n > max ( X 1 , ⋯ , X n - 1 ) . Trying to generalize this concept to random vectors, or even stochastic processes with continuous sample paths, we introduce two different concepts: A simple record is a stochastic process (or a random vector) $${\varvec{X}}_n$$ X n that is larger than $${\varvec{X}}_1,\cdots ,{\varvec{X}}_{n-1}$$ X 1 , ⋯ , X n - 1 in at least one component, whereas a complete record has to be larger than its predecessors in all components. In particular, the probability that a stochastic process $${\varvec{X}}_n$$ X n is a record as n tends to infinity is studied, assuming that the processes are in the max-domain of attraction of a max-stable process. Furthermore, the conditional distribution of $${\varvec{X}}_n$$ X n given that $${\varvec{X}}_n$$ X n is a record is derived.

Keywords: Champions and records; Multivariate extreme value distribution; Max-stable random vectors; D-norm; Max-stable processes; Max-domain of attraction; Primary; 60G70 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0811-7

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