A note on large deviations for unbounded observables

TL;DR

This paper investigates large deviations for unbounded observables in uniformly expanding systems, showing that uniform expansion alone does not guarantee a rate function and providing specific examples with various decay behaviors.

Contribution

It demonstrates that uniform expansion does not ensure a rate function for unbounded observables and provides detailed examples with different decay rates and behaviors.

Findings

Uniform expansion does not imply the existence of a rate function for unbounded observables.

Examples of unbounded observables with exponential decay of autocorrelations and stretched exponential large deviations.

A classical example of a bounded observable with no rate function is a degenerate coboundary.

Abstract

We consider exponential large deviations estimates for unbounded observables on uniformly expanding dynamical systems. We show that uniform expansion does not imply the existence of a rate function for unbounded observables no matter the tail behavior of the cumulative distribution function. We give examples of unbounded observables with exponential decay of autocorrelations, exponential decay under the transfer operator in each $L^p$, $1\le p < \infty$, and strictly stretched exponential large deviation. For observables of form $|\log d(x,p)|^{\alpha}$, $p$ periodic, on uniformly expanding systems we give the precise stretched exponential decay rate. We also show that a classical example in the literature of a bounded observable with exponential decay of autocorrelations yet with no rate function is degenerate as the observable is a coboundary.