Limit theory for some positive, stationary processes with infinite mean

TL;DR

This paper develops limit theorems and laws of the iterated logarithm for positive, stationary processes with infinite mean, extending Darling-Kac theory to infinite measure preserving transformations.

Contribution

It introduces new distributional limit theorems for a class of infinite mean processes, expanding the scope of existing ergodic theory.

Findings

Established distributional limit theorems for infinite mean processes

Proved one-sided laws of the iterated logarithm in this context

Extended Darling-Kac theory to new classes of transformations

Abstract

We prove distributional limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from non-integrable observables over certain piecewise expanding maps. This is done by extending Darling-Kac theory to a suitable family of infinite measure preserving transformations.