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Central limit theorems for sequential and random intermittent dynamical systems

Published online by Cambridge University Press:  22 September 2016

MATTHEW NICOL ,
ANDREW TÖRÖK  and
SANDRO VAIENTI
MATTHEW NICOL
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA email nicol@math.uh.edu
ANDREW TÖRÖK
Affiliation:
Department of Mathematics, University of Houston, Houston, TX 77204-3008, USA email nicol@math.uh.edu Institute of Mathematics of the Romanian Academy, PO Box 1-764, RO-70700 Bucharest, Romania email torok@math.uh.edu
SANDRO VAIENTI
Affiliation:
Aix Marseille Université, CNRS, CPT, UMR 7332, 13288 Marseille, France Université de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France email vaienti@cpt.univ-mrs.fr
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Abstract

We establish self-norming central limit theorems for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau–Manneville map. We also obtain quenched central limit theorems for random compositions of these maps.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 3 , May 2018 , pp. 1127 - 1153
Copyright
© Cambridge University Press, 2016 

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References

Aimino, R.. Vitesse de mélange et théorèmes limites pour les systèmes dynamiques aléatoires et non-autonomes. PhD Thesis, Université de Toulon, 2014.Google Scholar
Aimino, R., Hu, H., Nicol, M., Török, A. and Vaienti, S.. Polynomial loss of memory for maps of the interval with a neutral fixed point. Discrete Contin. Dyn. Syst. A 35(3) (2015), 793–806.Google Scholar
Aimino, R., Nicol, M. and Vaienti, S.. Annealed and quenched limit theorems for random expanding dynamical systems. Probab. Theory Related Fields 162(1) (2015), 233–274.Google Scholar
Ayyer, A., Liverani, C. and Stenlund, M.. Quenched CLT for random toral automorphism. Discrete Contin. Dyn. Syst. 24(2) (2009), 331–348.Google Scholar
Bahsoun, W. and Bose, C.. Mixing rates and limit theorems for random intermittent maps. Nonlinearity 29 (2016), 1417–1433.Google Scholar
Brown, B. M.. Martingale central limit theorems. Ann. Math. Statist. 42 (1971), 59–66.CrossRefGoogle Scholar
Conze, J.-P. and Raugi, A.. Limit theorems for sequential expanding dynamical systems on [0,1]. Ergodic Theory and Related Fields (Contemporary Mathematics, 430) . Ed. Assani, I.. American Mathematical Society, Providence, RI, 2007, pp. 89–121.CrossRefGoogle Scholar
Cuny, C. and Merlevède, F.. Strong invariance principles with rate for ‘reverse’ martingales and applications. J. Theoret. Probab. 28 (2015), 137–183.CrossRefGoogle Scholar
Dobbs, N. and Stenlund, M.. Quasistatic dynamical systems. Ergod. Th. & Dynam. Sys. doi:10.1017/etds.2016.9 , published online 12 May 2016.Google Scholar
Eagleson, G. K.. Some simple conditions for limit theorems to be mixing. Teor. Verojatnost. i Primenen. 21 (1976), 653–660. English translation: Theor. Prob. Appl. 21 (1976) 637–642, 1977.Google Scholar
Hall, P. and Heyde, C. C.. Martingale Limit Theory and its Application (Probability and Mathematical Statistics) . Academic Press, New York, 1980.Google Scholar
Haydn, N., Nicol, M., Persson, T. and Vaienti, S.. A note on Borel–Cantelli lemmas for non-uniformly hyperbolic dynamical systems. Ergod. Th. & Dynam. Sys. 33(2) (2013), 475–498.Google Scholar
Haydn, N., Nicol, M., Török, A. and Vaienti, S.. Almost sure invariance principle for sequential and non-stationary dynamical systems. Trans. Amer. Math. Soc. to appear, Preprint, 2014, arXiv:1406.4266.Google Scholar
Haydn, N., Nicol, M., Vaienti, S. and Zhang, L.. Central limit theorems for the shrinking target problem. J. Stat. Phys. 153 (2013), 864–887.Google Scholar
Leppänen, J. and Stenlund, M.. Quasistatic dynamics with intermittency. Math. Phys., Anal. Geom. 19 (2016), 1–23.Google Scholar
Liverani, C., Saussol, B. and Vaienti, S.. A probabilistic approach to intermittency. Ergod. Th. & Dynam. Sys. 19(3) (1999), 671–685.CrossRefGoogle Scholar
Peligrad, M.. Central limit theorem for triangular arrays of non-homogeneous Markov chains. Probab. Theory Related Fields 154 (2012), 409–428.Google Scholar
Schmidt, W.. A metrical theory in diophantine approximation. Canad. J. Math. 12 (1960), 619–631.Google Scholar
Schmidt, W.. Metrical theorems on fractional parts of sequences. Trans. Amer. Math. Soc. 110 (1964), 493–518.CrossRefGoogle Scholar
Sprindzuk, V. G.. Metric Theory of Diophantine Approximations (Scripta Series in Mathematics) . V. H. Winston and Sons, Washington, DC, 1979, translated from the Russian and edited by Richard A. Silverman, with a foreword by Donald J. Newman.Google Scholar