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Pré-Publication, Document De Travail Année : 2021

$\zeta$-functions and the topology of superlevel sets of stochastic processes

Résumé

We describe the topology of superlevel sets of (α-stable) Lévy processes X by introducing so-called stochastic ζ-functions, which are defined in terms of the widely used Pers pfunctional in the theory of persistence modules. The latter share many of the properties commonly attributed to ζ-functions in analytic number theory, among others, we show that for α-stable processes, these (tail) ζ-functions always admit a meromorphic extension to the entire complex plane with a single pole at α, of known residue and that the analytic properties of these ζ-functions are related to the asymptotic expansion of a dual variable, which counts the number of variations of X of size ≥ ε. Finally, using these results, we devise a new statistical parameter test using the topology of these superlevel sets.
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Dates et versions

hal-03372822 , version 1 (19-10-2021)

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  • HAL Id : hal-03372822 , version 1

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Daniel Perez. $\zeta$-functions and the topology of superlevel sets of stochastic processes. 2021. ⟨hal-03372822⟩
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