Dynamical attraction to stable processes
Fisher, Albert M. ; Talet, Marina
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012), p. 551-578 / Harvested from Numdam
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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.

Publié le : 2012-01-01
DOI : https://doi.org/10.1214/10-AIHP411
Classification:  37A50,  60F17,  60G18,  60G52 
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     title = {Dynamical attraction to stable processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {48},
     year = {2012},
     pages = {551-578},
     doi = {10.1214/10-AIHP411},
     mrnumber = {2954266},
     zbl = {1246.37020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2012__48_2_551_0}
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Fisher, Albert M.; Talet, Marina. Dynamical attraction to stable processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) pp. 551-578. doi : 10.1214/10-AIHP411. http://gdmltest.u-ga.fr/item/AIHPB_2012__48_2_551_0/

[1] J. Aaronson. An Introduction to Infinite Ergodic Theory. Mathematical Surveys and Monographs 50. Amer. Math. Soc., Providence, RI, 1997. | MR 1450400 | Zbl 0882.28013

[2] W. Ambrose and S. Kakutani. Structure and continuity of measurable flows. Duke Math. J. 9 (1942) 25-42. | MR 5800 | Zbl 0063.00065

[3] I. Berkes and H. Dehling. Some limit theorems in log density. Ann. Probab. 21 (1993) 1640-1670. | MR 1235433 | Zbl 0785.60014

[4] P. Billingsley. Convergence of Probability Measures. Wiley, New York, 1968. | MR 233396 | Zbl 0944.60003

[5] N. H. Bingham, C. M. Goldie and J. L. Teugels. Regular Variation. Cambridge Univ. Press, Cambridge/New York, 1987. | MR 898871 | Zbl 0667.26003

[6] L. Breiman. On the tail behavior of sums of independent random variables. Z. Wahrsch. Verw. Gebiete 9 (1967) 20-25. | MR 226707 | Zbl 0339.60050

[7] G. Brosamler. An almost everywhere Central Limit Theorem. Math. Proc. Cambridge Philos. Soc. 104 (1988) 561-574. | MR 957261 | Zbl 0668.60029

[8] M. Csörgo and P. Révész. Strong Approximations in Probability and Statistics. Academic Press, New York, 1981. | MR 666546 | Zbl 0539.60029

[9] W. Feller. Probability Theory, Vol. II. Wiley, New York/Chichester/Brisbane/Toronto, 1966. | MR 210154

[10] W. Feller. Probability Theory, Vol. II, 2nd edition. Wiley, New York/Chichester/Brisbane/Toronto, 1971. | MR 270403

[11] A. M. Fisher. A Pathwise Central Limit Theorem for random walks. Preprint, Univ. Goettingen, 1989. Ann. Probab. To appear.

[12] A. M. Fisher. Convex-invariant means and a pathwise central limit theorem. Adv. in Math. 63 (1987) 213-246. | MR 877784 | Zbl 0627.60034

[13] A. M. Fisher, A. Lopes and M. Talet. Self-similar returns in the transition from finite to infinite measure. Unpublished manuscript, 2010.

[14] A. M. Fisher and M. Talet. The self-similar dynamics of renewal processes. Electron. J. Probab. 16 (2011) 929-961. | MR 2801456 | Zbl 1225.60143

[15] S. Fomin. Finite invariant measures in the flows (Russian). Rec. Math. (Mat. Sbornik) N.S. 12 (1943) 99-108. | MR 9097 | Zbl 0063.01405

[16] H. Furstenberg. Recurrence in Ergodic Theory and Combinatorial Number Theory. Princeton Univ. Press, Princeton, 1981. | MR 603625 | Zbl 0459.28023

[17] E. Glasner. Ergodic Theory via Joinings. Mathematicall Surveys and Monographs 101. Amer. Math. Soc., Providence, RI, 2003. | MR 1958753 | Zbl 1038.37002

[18] P. Hall. A comedy of errors: The canonical form for a stable characteristic function. Bull. London Math. Soc. 13 (1981) 23-27. | MR 599635 | Zbl 0505.60022

[19] V. V. Kalashnikov. A complete metric in the space D[0, ∞). J. Math. Sci. 47 (1989) 2725-2730. | MR 1040101 | Zbl 0739.60002

[20] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer, Berlin/Heidelberg/New York, 1988. | MR 917065 | Zbl 0734.60060

[21] J. Komlos, P. Major and G. Tusnady. An approximation of the partial sums of independent RV's, and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131. | MR 375412 | Zbl 0308.60029

[22] J. Komlos, P. Major and G. Tusnady. An approximation of the partial sums of independent RV's, and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58. | MR 402883 | Zbl 0307.60045

[23] M. Lacey and W. Philipp. A note on the almost sure central limit theorem. Statist. Probab. Lett. 9 (1990) 201-205. | MR 1045184 | Zbl 0691.60016

[24] P. Lévy. Sur le développement en fraction continue d'un nombre choisi au hasard. Compositio Math. 3 (1936) 286-303. | JFM 62.0246.01 | Numdam | MR 1556945

[25] P. Major. Approximation of partial sums of i.i.d. r.v.s when the summands have only two moments. Z. Wahrsch. Verw. Gebiete 35 (1976) 221-229. | MR 415744 | Zbl 0338.60032

[26] P. Major. The approximation of partial sums of independent r.v.s. Z. Wahrsch. Verw. Gebiete 35 (1976) 213-220. | MR 415743 | Zbl 0338.60031

[27] P. Major. Almost sure functional limit theorems. Part I. The general case. Studia Sci. Math. Hungar. 34 (1998) 273-304. | MR 1645214 | Zbl 0921.60033

[28] P. Major. Almost sure functional limit theorems. Part II. The case of independent random variables. Studia Sci. Math. Hungar. 36 (2000) 231-273. | MR 1768228 | Zbl 0993.60020

[29] D. Ornstein. Ergodic Theory, Randomness and Dynamical Systems. Yale Mathematical Monographs 5. Yale Univ. Press, New Haven, 1973. | MR 447525 | Zbl 0296.28016

[30] V. A. Rohlin. On the fundamental ideas of measure theory. Mat. Sb. 25 (1949) 107-150. | MR 30584

[31] G. Samorodnitsky and M. Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York, 1994. | Zbl 0925.60027

[32] P. Schatte. On strong versions of the central limit theorem. Math. Nachr. 137 (1988) 249-256. | MR 968997 | Zbl 0661.60031

[33] P. Shields. The Theory of Bernoulli Shifts. Univ. Chicago Press, Chicago, 1973. | MR 442198 | Zbl 0308.28011

[34] C. Stone. Weak convergence of stochastic processes defined on semi-infinite time intervals. Proc. Amer. Math. Soc. 14 (1963) 694-696. | MR 153046 | Zbl 0116.35602

[35] V. Strassen. An invariance principle for the law of the iterated logarithm. Z. Wahrsch. Verw. Gebiete 3 (1964) 211-226. | MR 175194 | Zbl 0132.12903

[36] V. Strassen. Almost-sure behavior of sums of independent random variables and martingales. In Proc. 5th Berkeley Symp. Math. Stat. and Prob., Vol. 2 315-343. Univ. California Press, Berkeley, CA, 1965. | MR 214118 | Zbl 0201.49903

[37] P. Walters. An Introduction to Ergodic Theory. Springer, New York/Berlin, 1982. | MR 648108 | Zbl 0299.28012

[38] W. Whitt. Some useful functions for functional limit theorems. Math. Oper. Res. 5 (1980) 67-85. | MR 561155 | Zbl 0428.60010