Growth of normalizing sequences in limit theorems for conservative maps

TL;DR

This paper investigates the growth rates of normalizing sequences in limit theorems for conservative maps, showing polynomial and stretched exponential growth are possible, but exponential growth is not.

Contribution

It demonstrates the impossibility of nondegenerate limit theorems with exponential growth normalizations and constructs examples with stretched exponential growth.

Findings

Normalizing sequences cannot grow exponentially for nondegenerate limit theorems.

Examples exist where normalizing sequences grow like a stretched exponential with exponent close to 1.

Classical limit theorems typically involve polynomial or slowly varying normalizations.

Abstract

We consider normalizing sequences that can give rise to nondegenerate limittheorems for Birkhoff sums under the iteration of a conservative map. Mostclassical limit theorems involve normalizing sequences that are polynomial,possibly with an additional slowly varying factor. We show that, ingeneral, there can be no nondegenerate limit theorem with a normalizingsequence that grows exponentially, but that there are examples where itgrows like a stretched exponential, with an exponent arbitrarily close to 1.