Central limit theorems for the shrinking target problem

Abstract

Suppose $B_i:=B(p,r_i)$ are nested balls of radius $r_i$ about a point $p$ in a dynamical system $(T,X,\mu )$. The question of whether $T^{i}x\in B_i$ infinitely often (i. o.) for $\mu$ a.e.\ $x$ is often called the shrinking target problem. In many dynamical settings it has been shown that if $E_n:={\sum }_{i=1}^n\mu (B_i)$ diverges then there is a quantitative rate of entry and ${\lim }_{n\to \infty }\frac{1}{E_n}{\sum }_{j=1}^n{1}_{B_i}(T^{i}x)\to 1$ for $\mu$ a.e. $x\in X$. This is a self-norming type of strong law of large numbers. We establish self-norming central limit theorems (CLT) of the form ${\lim }_{n\to \infty }\frac{1}{a_n}{\sum }_{i=1}^n[1_{B_i}(T^{i}x)-\mu (B_i)]\to N(0,1)$ (in distribution) for a variety of hyperbolic and non-uniformly hyperbolic dynamical systems, the normalization constants are $a_n^2\sim E[{\sum }_{i=1}^n{1}_{B_i}(T^{i}x)-\mu (B_i){]}^2$. Dynamical systems to which our results apply include smooth expanding maps of the interval, Rychlik type maps, Gibbs-Markov maps, rational maps and, in higher dimensions, piecewise expanding maps. For such central limit theorems the main difficulty is to prove that the non-stationary variance has a limit in probability.

References in corpus (2)

• A Borel-Cantelli lemma for intermittent interval maps
• A central limit theorem for time-dependent dynamical systems

Cited by in corpus (5)

• Extremes and Recurrence in Dynamical Systems
• Annealed and quenched limit theorems for random expanding dynamical systems
• Central limit theorems for sequential and random intermittent dynamical systems
• Non-stationary Almost Sure Invariance Principle for Hyperbolic Systems with Singularities
• The Lindeberg theorem for Gibbs-Markov dynamics