Dynamical attraction to stable processes

Fisher, Albert M.; Talet, Marina

Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 2, pp. 551-578.

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.

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.

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