Darling-Kac theorem for renewal shifts in the absence of regular variation

TL;DR

This paper extends the Darling-Kac theorem to null recurrent renewal Markov chains with renewal distributions attracted to semistable laws, providing limit theorems along subsequences and exploring properties of the limit distributions.

Contribution

It introduces a new limit theorem for renewal shifts without regular variation, along subsequences, and characterizes the class of subsequences where the results hold.

Findings

Derived a Darling-Kac type limit theorem along subsequences.

Identified the class of subsequences for which the limit theorem applies.

Provided examples of infinite measure systems fitting the theory.

Abstract

We study null recurrent renewal Markov chains with renewal distribution in the domain of geometric partial attraction of a semistable law. Using the classical procedure of inversion, we derive a limit theorem similar to the Darling-Kac law along subsequences and obtain some interesting properties of the limit distribution. Also in this context, we obtain a Karamata type theorem along subsequences for positive operators. In both results, we identify the allowed class of subsequences. We provide several examples of nontrivial infinite measure preserving systems to which these results apply.