Nonconventional limit theorems in averaging
Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, p. 236-255

We consider “nonconventional” averaging setup in the form $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B\left({X}^{\epsilon }\left(t\right)$, $𝛯\left({q}_{1}\left(t\right)\right),𝛯\left({q}_{2}\left(t\right)\right),...,𝛯\left({q}_{\ell }\left(t\right)\right)\right)$ where $𝛯\left(t\right)$, $t\ge 0$ is either a stochastic process or a dynamical system with sufficiently fast mixing while ${q}_{j}\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}<{\alpha }_{2}<\cdots <{\alpha }_{k}$ and ${q}_{j}$, $j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the “nonconventional” averaging principle is asymptotically Gaussian.

Nous considérons un cadre non conventionnel de moyenne de la forme $\frac{\mathrm{d}{X}^{\epsilon }\left(t\right)}{\mathrm{d}t}=\epsilon B\left({X}^{\epsilon }\left(t\right)$, $𝛯\left({q}_{1}\left(t\right)\right),𝛯\left({q}_{2}\left(t\right)\right),...,𝛯\left({q}_{\ell }\left(t\right)\right)\right)$$𝛯\left(t\right)$, $t\ge 0$ est un processus stochastique ou un système dynamique suffisamment mélangeant tandis que ${q}_{j}\left(t\right)={\alpha }_{j}t$, ${\alpha }_{1}<{\alpha }_{2}<\cdots <{\alpha }_{k}$ et ${q}_{j}$, $j=k+1,...,\ell$ ont une croissance sur-linéaire. Nous montrons que le terme d’erreur après renormalisation est asymptotiquement gaussien.

DOI : https://doi.org/10.1214/12-AIHP514
Classification:  34C29,  60F17,  37D20
Keywords: averaging, limit theorems, martingales, hyperbolic dynamical systems
@article{AIHPB_2014__50_1_236_0,
author = {Kifer, Yuri},
title = {Nonconventional limit theorems in averaging},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
publisher = {Gauthier-Villars},
volume = {50},
number = {1},
year = {2014},
pages = {236-255},
doi = {10.1214/12-AIHP514},
mrnumber = {3161530},
language = {en},
url = {http://www.numdam.org/item/AIHPB_2014__50_1_236_0}
}

Kifer, Yuri. Nonconventional limit theorems in averaging. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 1, pp. 236-255. doi : 10.1214/12-AIHP514. http://www.numdam.org/item/AIHPB_2014__50_1_236_0/

 I. Assani. Multiple recurrence and almost sure convergence for weakly mixing dynamical systems. Israel J. Math. 103 (1998) 111-124. | MR 1613556 | Zbl 0920.28011

 V. Bergelson. Weakly mixing PET. Ergodic Theory Dynam. Systems 7 (1987) 337-349. | MR 912373 | Zbl 0645.28012

 R. Bowen. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. 470. Springer, Berlin, 1975. | MR 442989 | Zbl 0308.28010

 A. N. Borodin. A limit theorem for solutions of differential equations with random right-hand side. Theory Probab. Appl. 22 (1977) 482-497. | MR 517995 | Zbl 0412.60067

 R. C. Bradley. Introduction to Strong Mixing Conditions. Kendrick Press, Heber City, 2007. | Zbl 1134.60004

 V. Bergelson, A. Leibman and C. G. Moreira. From discrete-to continuous time ergodic theorems. Ergodic Theory Dynam. Systems. 32 (2012) 383-426. | MR 2901353 | Zbl 1251.37004

 D. Dolgopyat. On decay of correlations in Anosov flows. Ann. of Math. (2) 147 (1998) 357-390. | MR 1626749 | Zbl 0911.58029

 D. Dolgopyat. Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 (2003) 1637-1689. | MR 2034323 | Zbl 1031.37031

 D. Dolgopyat. Averaging and invariant measures. Mosc. Math. J. 5 (2005) 537-576. | MR 2241812 | Zbl pre05140621

 J. Doob. Stochastic Processes. Wiley, New York, 1953. | MR 58896 | Zbl 0696.60003

 D. Dolgopyat and C. Liverani. Energy transfer in a fast-slow Hamiltonian system. Comm. Math. Phys. 308 (2011) 201-225. | MR 2842975 | Zbl 1235.82065

 H. Furstenberg. Nonconventional ergodic averages. Proc. Sympos Pure Math. 50 (1990) 43-56. | MR 1067751 | Zbl 0711.28006

 M. Field, I. Melbourne and A. Torok. Decay of correlations, central limit theorems and approximation by Brownian motion for compact Lie group extensions. Ergodic Theory Dynam. Systems 23 (2003) 87-110. | MR 1971198 | Zbl 1140.37315

 M. Field, I. Melbourne and A. Torok. Stability of mixing and rapid mixing for hyperbolic flows. Ann. of Math. (2) 166 (2007) 269-291. | MR 2342697 | Zbl 1140.37004

 L. Heinrich. Mixing properties and central limit theorem for a class of non-identical piecewise monotonic ${C}^{2}$-transformations. Math. Nachr. 181 (1996) 185-214. | MR 1409076 | Zbl 0863.60023

 I. A. Ibragimov and Y. V. Linnik. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen, 1971. | MR 322926 | Zbl 0219.60027

 J. Jacod and A. N. Shiryaev. Limit Theorems for Stochastic Processes, 2nd edition. Springer, Berlin, 2003. | MR 1943877 | Zbl 0635.60021

 R. Z. Khasminskii. On stochastic processes defined by differential equations with a small parameter. Theory Probab. Appl. 11 (1966) 211-228. | MR 203788 | Zbl 0168.16002

 R. Z. Khasminskii. A limit theorem for solutions of differential equations with random right-hand side. Theory Probab. Appl. 11 (1966) 390-406. | Zbl 0202.48601

 Yu. Kifer. Limit theorems in averaging for dynamical systems. Ergodic Theory Dynam. Systems 15 (1995) 1143-1172. | MR 1366312 | Zbl 0841.34048

 Yu. Kifer. Averaging principle for fully coupled dynamical systems and large deviations. Ergodic Theory Dynam. Systems 24 (2004) 847-871. | MR 2062922 | Zbl 1055.37025

 Y. Kifer. Nonconventional law of large numbers and fractal dimensions of some multiple recurrence sets. Stoch. Dyn. 12 (2012) 1150023. | MR 2926580 | Zbl 1255.60044

 Y. Kifer. A strong invariance principle for nonconventional sums. Probab. Theory Related Fields 155(1-2) (2013) 463-486. | MR 3010405 | Zbl 1271.60047

 A. Katok and B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge Univ. Press, Cambridge, 1995. | MR 1326374 | Zbl 0878.58019

 Yu. Kifer and S. R. S. Varadhan. Nonconventional limit theorems in discrete and continuous time via martingales. Ann. Probab. To appear. | MR 3178470 | Zbl pre06288290

 C. Liverani. Central limit theorems for deterministic systems. In International Conference on Dynamical Systems (Montevideo, 1995) 56-75. Pitman Research Notes in Math. 363. Longman, Harlow, 1996. | MR 1460797 | Zbl 0871.58055

 D. L. Mcleish. Invariance principles for dependent variables. Z. Wahrsch. Verw. Gebiete 32 (1975) 165-178. | MR 388483 | Zbl 0288.60034

 D. L. Mcleish. On the invariance principle for nonstationary mixingales. Ann. Probab. 5 (1977) 616-621. | MR 445583 | Zbl 0367.60021

 J. A. Sanders, F. Verhurst and J. Murdock. Averaging Methods in Nonlinear Dynamical Systems, 2nd edition. Springer, New York, 2007. | MR 2316999 | Zbl 1128.34001