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Almost surely invariance principle for non-stationary and random intermittent dynamical systems
Department of Mathematics, University of Houston, Houston, Texas 77204-3008, USA |
We establish almost sure invariance principles (ASIP), a strong form of approximation by Brownian motion, 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. Quenched ASIP for random compositions of these maps is also obtained.
References:
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R. Aimino, H. Y. Hu, M. Nicol, A. Török and S. Vaienti,
Polynomial loss of memory for maps of the interval with a neutral fixed point, Discrete Contin. Dyn. Syst., 35 (2015), 793-806.
doi: 10.3934/dcds.2015.35.793. |
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C. Cuny and F. Merlevède,
Strong invariance principles with rate for "reverse" martingale differences and applications, J. Theoret. Probab., 28 (2015), 137-183.
doi: 10.1007/s10959-013-0506-z. |
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D. Dragičević, G. Froyland, C. González-Tokman and S. Vaienti,
Almost sure invariance principle for random piecewise expanding maps, Nonlinearity, 31 (2018), 2252-2280.
doi: 10.1088/1361-6544/aaaf4b. |
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N. Haydn, M. Nicol, A. Török and S. Vaienti,
Almost sure invariance principle for sequential and non-stationary dynamical systems, Trans. Amer. Math. Soc., 369 (2017), 5293-5316.
doi: 10.1090/tran/6812. |
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O. Hella and J. Leppänen, Central limit theorems with a rate of convergence for time-dependent intermittent maps, arXiv E-Prints, arXiv: 1811.11170. |
| [6] |
M. Nicol, A. Török and S. Vaienti,
Central limit theorems for sequential and random intermittent dynamical systems, Ergodic Theory Dynam. Systems, 38 (2018), 1127-1153.
doi: 10.1017/etds.2016.69. |
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D. J. Scott and R. M. Huggins,
On the embedding of processes in Brownian motion and the law of the iterated logarithm for reverse martingales, Bull. Austral. Math. Soc., 27 (1983), 443-459.
doi: 10.1017/S0004972700025946. |
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V. G. Sprindžuk, Metric Theory of Diophantine Approximations, Translated from the Russian and edited by Richard A. Silverman, With a foreword by Donald J. Newman, Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D. C., A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979. |
show all references
References:
| [1] |
R. Aimino, H. Y. Hu, M. Nicol, A. Török and S. Vaienti,
Polynomial loss of memory for maps of the interval with a neutral fixed point, Discrete Contin. Dyn. Syst., 35 (2015), 793-806.
doi: 10.3934/dcds.2015.35.793. |
| [2] |
C. Cuny and F. Merlevède,
Strong invariance principles with rate for "reverse" martingale differences and applications, J. Theoret. Probab., 28 (2015), 137-183.
doi: 10.1007/s10959-013-0506-z. |
| [3] |
D. Dragičević, G. Froyland, C. González-Tokman and S. Vaienti,
Almost sure invariance principle for random piecewise expanding maps, Nonlinearity, 31 (2018), 2252-2280.
doi: 10.1088/1361-6544/aaaf4b. |
| [4] |
N. Haydn, M. Nicol, A. Török and S. Vaienti,
Almost sure invariance principle for sequential and non-stationary dynamical systems, Trans. Amer. Math. Soc., 369 (2017), 5293-5316.
doi: 10.1090/tran/6812. |
| [5] |
O. Hella and J. Leppänen, Central limit theorems with a rate of convergence for time-dependent intermittent maps, arXiv E-Prints, arXiv: 1811.11170. |
| [6] |
M. Nicol, A. Török and S. Vaienti,
Central limit theorems for sequential and random intermittent dynamical systems, Ergodic Theory Dynam. Systems, 38 (2018), 1127-1153.
doi: 10.1017/etds.2016.69. |
| [7] |
D. J. Scott and R. M. Huggins,
On the embedding of processes in Brownian motion and the law of the iterated logarithm for reverse martingales, Bull. Austral. Math. Soc., 27 (1983), 443-459.
doi: 10.1017/S0004972700025946. |
| [8] |
V. G. Sprindžuk, Metric Theory of Diophantine Approximations, Translated from the Russian and edited by Richard A. Silverman, With a foreword by Donald J. Newman, Scripta Series in Mathematics. V. H. Winston & Sons, Washington, D. C., A Halsted Press Book, John Wiley & Sons, New York-Toronto, Ont.-London, 1979. |
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