Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Dynamical attraction to stable processes

Albert M. Fisher and Marina Talet

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Abstract

We apply dynamical ideas within probability theory, proving an almost-sure invariance principle in log density for stable processes. The familiar scaling property (self-similarity) of the stable process has a stronger expression, that the scaling flow on Skorokhod path space is a Bernoulli flow. We prove that typical paths of a random walk with i.i.d. increments in the domain of attraction of a stable law can be paired with paths of a stable process so that, after applying a non-random regularly varying time change to the walk, the two paths are forward asymptotic in the flow except for a set of times of density zero. This implies that a.e. time-changed random walk path is a generic point for the flow, i.e. it gives all the expected time averages. For the Brownian case, making use of known results in the literature, one has a stronger statement: the random walk and the Brownian paths are forward asymptotic under the scaling flow (now with no exceptional set of times), at an exponential rate given by the moment assumption.

Résumé

En appliquant des idées venues des systèmes dynamiques aux probabilités, nous prouvons un principe d’invariance presque sûr au sens de la densité logarithmique pour des processus stables. L’auto-similarité d’un processus stable revêt une expression plus forte, celle de la Bernoullicité du flot d’échelle agissant sur l’espace de Skorokhod des trajectoires. Nous montrons qu’il existe un couplage de la marche aléatoire à accroissements i.i.d. dans le domaine d’attraction d’une loi stable et d’un processus stable tel que presque sûrement, après un changement de temps déterministe et à variation régulière, sous l’action du flot d’échelle, les deux processus soient asymptotiques dans le futur sauf pour un ensemble de temps de densité nulle. Il en découle que presque toute marche (à un changement de temps près) est un point générique du flot. Dans le cas brownien, compte-tenu de résultats bien connus dans la littérature, nous avons un résultat plus fort : sous l’action du flot, les trajectoires de la marche et du brownien sont asymptotiques dans le futur avec une vitesse exponentielle donnée par l’hypothèse de moment.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 2 (2012), 551-578.

Dates
First available in Project Euclid: 11 April 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1334148210

Digital Object Identifier
doi:10.1214/10-AIHP411

Mathematical Reviews number (MathSciNet)
MR2954266

Zentralblatt MATH identifier
1246.37020

Subjects
Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10] 60F17: Functional limit theorems; invariance principles 60G18: Self-similar processes 60G52: Stable processes

Keywords
Brownian motion Stable process Almost-sure invariance principle in log density Generic point Pathwise Central Limit Theorem Scaling flow

Citation

Fisher, Albert M.; Talet, Marina. Dynamical attraction to stable processes. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 2, 551--578. doi:10.1214/10-AIHP411. https://projecteuclid.org/euclid.aihp/1334148210


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