Biharmonic functions on groups and limit theorems for quasimorphisms along random walks

TL;DR

This paper establishes that quasimorphisms on groups can be approximated by quasi-biharmonic functions, enabling the derivation of limit theorems for random walks on groups using martingale techniques.

Contribution

It introduces a general approximation of quasimorphisms by quasi-biharmonic functions and applies classical martingale theorems to derive limit laws for random walks.

Findings

Non-degenerate central limit theorems for quasimorphisms

Laws of the iterated logarithm for quasimorphisms

Integral representations via martingale convergence

Abstract

We show for very general classes of measures on locally compact second countable groups that every Borel measurable quasimorphism is at bounded distance from a quasi-biharmonic one. This allows us to deduce non-degenerate central limit theorems and laws of the iterated logarithm for such quasimorphisms along regular random walks on topological groups using classical martingale limit theorems of Billingsley and Stout. For quasi-biharmonic quasimorphism on countable groups we also obtain integral representations using martingale convergence.