Some almost sure results for unbounded functions of intermittent maps and their associated Markov chains

TL;DR

This paper investigates the statistical properties of unbounded functions of certain intermittent maps and their Markov chains, establishing limit theorems and invariance principles for their partial sums.

Contribution

It introduces a broad class of unbounded functions for which classical limit theorems hold in the context of intermittent maps and associated Markov chains.

Findings

Central limit theorem and law of the iterated logarithm for partial sums of f∘T^i

Strong invariance principle for partial sums of f(Y_i)

Almost sure convergence rates matching i.i.d. case for stable law domains

Abstract

We consider a large class of piecewise expanding maps T of [0,1] with a neutral fixed point, and their associated Markov chain Y_i whose transition kernel is the Perron-Frobenius operator of T with respect to the absolutely continuous invariant probability measure. We give a large class of unbounded functions f for which the partial sums of f\circ T^i satisfy both a central limit theorem and a bounded law of the iterated logarithm. For the same class, we prove that the partial sums of f(Y_i) satisfy a strong invariance principle. When the class is larger, so that the partial sums of f\circ T^i may belong to the domain of normal attraction of a stable law of index p\in (1, 2), we show that the almost sure rates of convergence in the strong law of large numbers are the same as in the corresponding i.i.d. case.