The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic (Anosov) dynamics developed by the first author et al. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for vector-valued observables. We stress that the previous works considered exclusively the case of scalar-valued observables. In another direction, we show that this method can be used to establish a variety of new limit laws (either for scalar or vector-valued observables) that have not been discussed previously in the literature for the classes of dynamics we consider. More precisely, we establish the moderate deviations principle, concentration inequalities, Berry–Esseen estimates as well as Edgeworth and large deviation expansions. Although our techniques rely on the approach developed in the previous works of the first author et al., we emphasize that our arguments require several nontrivial adjustments as well as new ideas.

本文的目的是双重的。在一个方向上，我们扩展了由第一作者等人开发的用于随机分段扩展和双曲线（Anosov）动力学的频谱方法。为向量值建立大偏差原理，中心极限定理和局部中心极限定理的淬火形式可观察的。我们强调指出，先前的工作仅考虑标量值可观测对象的情况。在另一个方向上，我们表明该方法可用于建立各种新的极限定律（用于标量或矢量值可观测值），这些新的极限定律在文献中之前并未针对我们考虑的动力学类别进行过讨论。更准确地说，我们建立了中等偏差原理，浓度不等式，Berry-Esseen估计以及Edgeworth和大偏差展开。尽管我们的技术依赖于第一作者等人先前工作中发展的方法，但我们强调，我们的论点需要进行一些不平凡的调整以及新思想。