Generalized Fibonacci numbers and extreme value laws for the Rényi map

N. B. Boer, A. E. Sterk*

*Bijbehorende auteur voor dit werk

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In this paper we prove an extreme value law for a stochastic process obtained by iterating the Rényi map x↦βx(mod1), where we assume that β>1 is an integer. Haiman (2018) derived a recursion formula for the Lebesgue measure of threshold exceedance sets. We show how this recursion formula is related to a rescaled version of the k-generalized Fibonacci sequence. For the latter sequence we derive a Binet formula which leads to a closed-form expression for the distribution of partial maxima of the stochastic process. The proof of the extreme value law is completed by deriving sharp bounds for the dominant root of the characteristic polynomial associated with the Fibonacci sequence.

Originele taal-2English
Pagina's (van-tot)704-718
Aantal pagina's15
TijdschriftIndagationes Mathematicae
Volume32
Nummer van het tijdschrift3
DOI's
StatusPublished - mei-2021

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