Chapitre D'ouvrage Année : 2025

An application of Sparre Andersen's fluctuation theorem for exchangeable and sign-invariant random variables

Résumé

We revisit here a famous result by Sparre Andersen on persistence probabilities $\mathbf{P}(S_k\geq 0 \;\forall\, 0\leq k\leq n)$ for symmetric random walks $(S_n)_{n\geq 0}$. We give a short proof of this result when considering sums of random variables that are only assumed exchangeable and sign-invariant. We then apply this result to the study of persistence probabilities of (symmetric) additive functionals of Markov chains, which can be seen as a natural generalization of integrated random walks

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Dates et versions

hal-04075157 , version 1 (19-04-2023)
hal-04075157 , version 2 (05-09-2025)

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Quentin Berger, Loïc Béthencourt. An application of Sparre Andersen's fluctuation theorem for exchangeable and sign-invariant random variables. Séminaire de Probabilités LII, LII, Springer Nature Switzerland, pp 367-388, 2025, Lecture Notes in Mathematics, 978-3-031-86421-6. ⟨10.1007/978-3-031-86422-3_9⟩. ⟨hal-04075157v2⟩
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