The almost sure invariance principle for unbounded functions of expanding maps

TL;DR

This paper proves an almost sure invariance principle for unbounded functions of certain expanding maps, extending classical results to broader classes of functions and maps with neutral fixed points.

Contribution

It establishes the almost sure invariance principle for unbounded functions in two classes of expanding maps, including those with neutral fixed points, under new integrability and tail conditions.

Findings

Valid for square integrable functions in uniformly expanding maps.

Applicable to functions with specific tail decay in maps with neutral fixed points.

Extends invariance principles to unbounded functions beyond bounded observables.

Abstract

We consider two classes of piecewise expanding maps $T$ of $[0,1]$: a class of uniformly expanding maps for which the Perron-Frobenius operator has a spectral gap in the space of bounded variation functions, and a class of expanding maps with a neutral fixed point at zero. In both cases, we give a large class of unbounded functions $f$ for which the partial sums of $f\circ T^i$ satisfy an almost sure invariance principle. This class contains piecewise monotonic functions (with a finite number of branches) such that: - For uniformly expanding maps, they are square integrable with respect to the absolutely continuous invariant probability measure. - For maps having a neutral fixed point at zero, they satisfy an (optimal) tail condition with respect to the absolutely continuous invariant probability measure.