An analogue of Law of Iterated Logarithm for Heavy Tailed Random Variables

TL;DR

This paper proves a functional law of the iterated logarithm for heavy-tailed observables in expanding circle maps, revealing the limit behavior of sums with infinite variance using novel techniques.

Contribution

It introduces a new functional limit theorem for heavy-tailed processes, extending classical results to cases with infinite variance.

Findings

Limit points are piecewise constant functions with finitely many jumps.

Rescaled sums converge to a set of increasing step functions.

The approach combines trimming and Borel-Cantelli techniques.

Abstract

We establish functional limit theorems for ergodic sums of observables with power singularities for expanding circle maps. In the regime where the observables have infinite variance, we show that when rescaled by $N^{1/s}(\ln N)^\alpha$, the partial sum process has limit points consisting precisely of increasing piecewise constant functions with finitely many jumps. Our approach combines trimming techniques with a multiple Borel-Cantelli argument. It provides a functional law of the iterated logarithm for heavy-tailed processes where classical almost sure invariance principles do not apply.