Central limit theorems with a rate of convergence for time-dependent intermittent maps

Abstract

We study dynamical systems arising as time-dependent compositions of Pomeau-Manneville-type intermittent maps. We establish central limit theorems for appropriately scaled and centered Birkhoff-like partial sums, with estimates on the rate of convergence. For maps chosen from a certain parameter range, but without additional assumptions on how the maps vary with time, we obtain a self-norming CLT provided that the variances of the partial sums grow sufficiently fast. When the maps are chosen randomly according to a shift-invariant probability measure, we identify conditions under which the quenched CLT holds, assuming fiberwise centering. Finally, we show a multivariate CLT for intermittent quasistatic systems. Our approach is based on Stein's method of normal approximation.Acknowledgements. The authors would like to thank Mikko Stenlund for helpful comments on preliminary versions of the manuscript.

Large deviations and central limit theorems for sequential and random systems of intermittent maps

“…• We show that the sequential CLT in [NTV18, Theorem 3.1] and [HL20] holds for the sharp α < 1 2 (from α < 1 9 ) if the variance grows at the rate specified. • We show that the CLT holds not only with respect to Lebesgue measure m but also for d m = x −α dm, which scales at the origin as the invariant measure of T α .…”

“…Various scenarios under which σ 2 n (ω) > n β are given in [NTV18]. See also [HL20]. If the maps T ω i preserve the same invariant measure, then it suffices to consider observables with mean zero, since the mean would be the same along each realization.…”

Quenched Normal Approximation for Random Sequences of Transformations

“…1-Lipschitz) one has d(σ n Z, σZ) ≤ C|σ 2 − σ 2 n | where Z ∼ N (0, I d×d ) and C = C(σ), which again yields an estimate of the type d(W n , σZ) ≤ d(W n , σ n Z) + C|σ 2 − σ 2 n |. We refer the reader to Hella [13] for details, including the hard part of establishing an almost sure, asymptotically decaying bound on d(W n , σ n Z) in the vector-valued case.…”

Almost surely invariance principle for non-stationary and random intermittent dynamical systems

“…However, if the transfer operator with respect to the Lebesgue measure is not quasi-compact on a suitable Banach space, the approach described above fails to work. Such example for non-stationary dynamical systems and related statistical properties are provided in a preceding series of papers [1,5,6].…”

“…The last two papers [5,6] considered the same system and proved self-norming CLT under the assumption that it is sufficiently chaotic and the variance grows at a certain rate. Moreover, they proved self-norming CLT for nearby maps and quenched CLT for random compositions of maps in the same family provided the system is sufficiently chaotic and the base map of random dynamical system has strong mixing.…”