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Article Dans Une Revue Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques Année : 2012

Dynamical attraction to stable processes

Résumé

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.

Dates et versions

hal-01285508 , version 1 (09-03-2016)

Identifiants

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Marina Talet, Albert Meads Fisher. Dynamical attraction to stable processes. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2012, 48 (2), pp.551-578. ⟨10.1214/10-AIHP411⟩. ⟨hal-01285508⟩
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