Moment bounds and concentration inequalities for slowly mixing dynamical systems

Abstract

We obtain optimal moment bounds for Birkhoff sums, and optimal concentration inequalities, for a large class of slowly mixing dynamical systems, including those that admit anomalous diffusion in the form of a stable law or a central limit theorem with nonstandard scaling $(n\log n)^{1/2}$

Large and moderate deviations for bounded functions of slowly mixing Markov chains

“…So, provided that p < 1/(1 − α), one can take 2p > r > [DM15]). We refer the interested readers for instance to the introduction of [GM14], where motivations, examples and definitions are given. Our theorem applies to such examples, and improves the previous upper bounds of the literature such as [Mel09] who obtained, when α ∈ (1/2, 1) and p ≥ 2, a rate of order (ln n) 1−p n (p−1)(2α−1) instead of n αp−1 in (1.2).…”

“…In this section, we will for simplicity formulate the results for the dynamics, as the estimates of [GM14] we will use are formulated in this context. The class of functionals for which we will prove moderate deviations is the class of separately Lipschitz functions: these are the functions K = K(z 0 , .…”

“…We just follow the lines of the proof of Lemma 6.2 in [CG12] up to (6.1). Note that this paper requires the condition p > 2 (for the validity of (4.8) there), but Lemma 4.2 in [GM14] replaces this inequality for p = 2.…”

“…Its average under π vanishes. The corresponding Markov chain satisfies H(p) for the Hölder norm on the tower, see for instance [GM14] and references therein.…”

Measuring logarithmic corrections to normal diffusion in infinite-horizon billiards

“…If the convergence in (3) were sufficiently strong, then the qth moment of the rescaled displacement vector distribution would converge to M q for all q. However, the weak convergence (3) and a known bound on the second moment [18,42], viz. r 2 (t) K t log t, imply this convergence of the moments only for q < 2 [43, Sec.…”