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On Functional Records and Champions

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Abstract

A record among a sequence of iid random variables \(X_1,X_2,\dots \) on the real line is defined as a member \(X_n\) such that \(X_n>\max (X_1,\cdots ,X_{n-1})\). 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) \({\varvec{X}}_n\) that is larger than \({\varvec{X}}_1,\cdots ,{\varvec{X}}_{n-1}\) 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 \({\varvec{X}}_n\) 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 \({\varvec{X}}_n\) given that \({\varvec{X}}_n\) is a record is derived.

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Correspondence to Michael Falk.

Additional information

This work was supported by a Research Grant (VKR023480) for Michael Falk from VILLUM FONDEN. This work was supported by a Research Grant (91587278) for Maximilian Zott from the German Academic Exchange Service (DAAD)

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Dombry, C., Falk, M. & Zott, M. On Functional Records and Champions. J Theor Probab 32, 1252–1277 (2019). https://doi.org/10.1007/s10959-018-0811-7

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