Dynamical attraction to stable processes

Albert M. Fisher; Marina Talet

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 2, page 551-578
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Fisher, Albert M., and Talet, Marina. "Dynamical attraction to stable processes." Annales de l'I.H.P. Probabilités et statistiques 48.2 (2012): 551-578. <http://eudml.org/doc/272041>.

@article{Fisher2012,
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.},
author = {Fisher, Albert M., Talet, Marina},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {brownian motion; stable process; almost-sure invariance principle in log density; generic point; pathwise central limit theorem; scaling flow; Brownian motion},
language = {eng},
number = {2},
pages = {551-578},
publisher = {Gauthier-Villars},
title = {Dynamical attraction to stable processes},
url = {http://eudml.org/doc/272041},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Fisher, Albert M.
AU - Talet, Marina
TI - Dynamical attraction to stable processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 2
SP - 551
EP - 578
AB - 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.
LA - eng
KW - brownian motion; stable process; almost-sure invariance principle in log density; generic point; pathwise central limit theorem; scaling flow; Brownian motion
UR - http://eudml.org/doc/272041
ER -

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