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Dynamical attraction to stable processes. (English. French summary) Zbl 1246.37020
Authors’ 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.

MSC:
37A50 Dynamical systems and their relations with probability theory and stochastic processes
60F17 Functional limit theorems; invariance principles
60G18 Self-similar stochastic processes
60G52 Stable stochastic processes
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