Decay of correlations and limit theorems for random intermittent maps
Davor Dragicevic, Yeor Hafouta, Juho Leppanen
TL;DR
This work analyzes random intermittent maps of LSV type under time-dependent, mixing environments to establish quenched and annealed probabilistic limit theorems. The authors introduce a rigorous abstract $p$-control framework to handle polynomial memory loss caused by a neutral fixed point, deriving CLTs, moment bounds, and ASIP for random $T_\omega$ and the associated transfer operators, with a key restriction $\gamma=\mathrm{essinf}(\beta) < \tfrac{1}{5}$ and sufficiently fast environmental mixing. They further obtain CLT rates and moment inequalities, and extend the analysis to annealed limits and to quenched memory loss for random LSV maps, including settings with $\alpha$-mixing noise. The results generalize previous iid or perturbative findings, give new ASIP and rate results under broad mixing hypotheses, and suggest applicability to random Young-tower-type settings in non-stationary environments.
Abstract
In this paper, we revisit the problem of polynomial memory loss and the central limit theorem for time-dependent LSV maps. More precisely, we show that for random LSV maps corresponding to a random parameter beta() we obtain quenched memory loss, decay of correlations, central limit theorems with rates, moment bounds and almost sure invariance principles (ASIP) when the essential infimum of beta() is less than 1/5 and the driving process (i.e. random environment) is mixing sufficiently fast. In [59, Corollary 3.8] the ASIP was obtained for ergodic driving systems when the essential supremum of \b{eta} is less than 1/2. As will be elaborated in Section 1, restrictions on the essential infimum are more natural in our context. Our results have an abstract form which we believe could be useful in other circumstances, as will be elaborated in a future work