Functional correlation decay and multivariate normal approximation for non-uniformly expanding maps

Abstract

ABSTRACT. In the setting of intermittent Pomeau-Manneville maps with time dependent parameters, we show a functional correlation bound widely useful for the analysis of the statistical properties of the model. We give two applications of this result, by showing that in a suitable range of parameters the bound implies the conditions of the normal approximation methods of Stein and Rio. For a single Pomeau-Manneville map belonging to this parameter range, both methods then yield a multivariate central limit theorem with a rate of convergence.

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

“…where the last inequality holds because L α is an L 1 -contraction. By Lemma 2.4 in [29], for each l with m + 1 ≤ l ≤ m + k, there are functions g 1 , g 2 ∈ C * (β * ), such that L τ (l−1)/n · · · L τ (m+1)/n gf p α = g 1 − g 2 , and g i 1 ≤ C(β * )( f Lip +1) for some constant C(β * ) > 0 depending only on T β * . Theorem 5.1 in [31] applies to cone functions, and it follows that…”

“…The study of time-dependent non-uniformly expanding maps was initiated in [2], where a statistical memory loss result was established for compositions of intermittent maps T αn with parameters 0 < α n ≤ β * < 1, but without any other assumptions on how the sequence (α n ) is chosen. The result was a key ingredient in the proof for the functional correlation bound of [29] -one of the main tools of the present manuscript -and it was also instrumental in [35], where the CLT was first examined in this setup. The main result of [35] shows that, in a certain parameter range, a CLT of the form (1) holds for all C 1 -observables f : [0, 1] → R, when µ is the Lebesgue measure, assuming again that Var µ (S) grows sufficiently fast.…”

“…(I) Since f is Lipschitz continuous and β * < 1/3, it follows from [29] (or from Theorem 3.3 below) that condition (I) holds for any ϕ > 2. (II) Let ν be a measure with density g ∈ C * (β * ).…”

“…This paper is a continuation of the previous works [21,23,29] which dealt with statistical properties of time-dependent dynamical systems, such as those described by compositions of the form T n • · · · • T 1 where each T n : X → X is a self-map of a probability space (X, B, µ). In [29], the constituent maps T n were chosen to be Pomeau-Manneville-type intermittent maps T αn : [0, 1] → [0, 1] with parameters α n ∈ (0, 1). The maps T αn are expanding but have a common neutral fixed point at the origin, whose influence becomes more significant as α n increases.…”

“…The purpose of the present paper is to extend the three applications described above to the setting considered in [29], i.e. when the time-dependent system is composed of non-uniformly expanding intermittent maps.…”