Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon

Quentin Berger; Matthias Birkner; Linglong Yuan

doi:10.5802/afst.1802

Quentin Berger ¹ ; Matthias Birkner ² ; Linglong Yuan ³

¹ Université Sorbonne Paris Nord, Laboratoire d’Analyse, Géométrie et Applications, CNRS UMR 7539. 99 Av. J-B Clément, 93430 Villetaneuse, France & Institut Universitaire de France, France
² Johannes-Gutenberg-Universität Mainz, Institut für Mathematik. Staudingerweg 9, 55128 Mainz, Germany
³ University of Liverpool, Department of Mathematical Sciences. Peach Street, L69 7ZL, Liverpool, UK

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 5, pp. 1413-1485.

Résumé
Abstract

Nous étudions dans cet article les grandes déviations et les grandes déviations locales pour des sommes de variables aléatoires réelles i.i.d. dans le domaine d’attraction d’une loi $α$ -stable, $α \in (0, 2]$ , en mettant l’accent sur le cas $α = 2$ . Il y a deux scénarios différents : soit la déviation est réalisée par un comportement collectif où toutes les variables sommées contribuent à la déviation (un scénario gaussien), ou bien une seule des variables est atypiquement grande et contribue à la déviation (un scénario d’un unique grand saut). De tels résultats sont connus quand $α \in (0, 2)$ (les grandes déviations suivent toujours le scénario d’un grand saut) ou quand les variables admettent un moment d’ordre $2 + δ$ pour un $δ > 0$ . Nous étendons ces résultats, en incluant en particulier le cas où la distribution possède une queue à droite à variation régulière d’indice $- 2$ (traitant le cas de variables de variance infinie dans le domaine d’attraction de la loi normale). Nous identifions le seuil pour la transition entre le régime gaussien et le régime de grand saut ; ce seuil est légèrement plus grand lorsque l’on considère les grandes déviations locales comparées aux grandes déviations intégrées. De plus, nous complétons nos résultats en décrivant le comportement de la somme et de la plus grande variable sommée conditionnellement à avoir une grande déviation (ou une grande déviation locale), pour tout $α \in (0, 2]$ , à la fois dans le régime gaussien et dans le régime d’un grand saut. Comme application, nous montrons comment nos résultats peuvent être utilisés pour étudier le phénomène de condensation dans le processus de rang zéro (zero-range process) de densité critique, étendant le spectre des paramètres précédemment considérés dans la littérature.

This article studies large and local large deviations for sums of i.i.d. real-valued random variables in the domain of attraction of an $α$ -stable law, $α \in (0, 2]$ , with emphasis on the case $α = 2$ . There are two different scenarios: either the deviation is realised via a collective behaviour with all summands contributing to the deviation (a Gaussian scenario), or a single summand is atypically large and contributes to the deviation (a one-big-jump scenario). Such results are known when $α \in (0, 2)$ (large deviations always follow a one big-jump scenario) or when the random variables admit a moment of order $2 + δ$ for some $δ > 0$ . We extend these results, including in particular the case where the right tail is regularly varying with index $- 2$ (treating cases with infinite variance in the domain of attraction of the normal law). We identify the threshold for the transition between the Gaussian and the one-big-jump regimes; it is slightly larger when considering local large deviations compared to integral large deviations. Additionally, we complement our results by describing the behaviour of the sum and of the largest summand conditionally on a (local) large deviation, for any $α \in (0, 2]$ , both in the Gaussian and in the one-big-jump regimes. As an application, we show how our results can be used in the study of condensation phenomenon in the zero-range process at the critical density, extending the range of parameters previously considered in the literature.

Reçu le : 2023-03-31
Accepté le : 2023-10-23
Publié le : 2025-04-24

DOI : 10.5802/afst.1802

Classification : 60F10, 60G50
Mots-clés : Large deviation, Local large deviation, Extended & intermediate regular variation, Phase transition, One-big-jump phenomenon, Condensation, $\alpha $-stable law

Affiliations des auteurs :

Quentin Berger ¹ ; Matthias Birkner ² ; Linglong Yuan ³

¹ Université Sorbonne Paris Nord, Laboratoire d’Analyse, Géométrie et Applications, CNRS UMR 7539. 99 Av. J-B Clément, 93430 Villetaneuse, France & Institut Universitaire de France, France
² Johannes-Gutenberg-Universität Mainz, Institut für Mathematik. Staudingerweg 9, 55128 Mainz, Germany
³ University of Liverpool, Department of Mathematical Sciences. Peach Street, L69 7ZL, Liverpool, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1413--1485},
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Quentin Berger; Matthias Birkner; Linglong Yuan. Collective vs. individual behaviour for sums of i.i.d. random variables: appearance of the one-big-jump phenomenon. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 5, pp. 1413-1485. doi : 10.5802/afst.1802. https://afst.centre-mersenne.org/articles/10.5802/afst.1802/

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