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We show for very general classes of measures on locally compact secondcountable groups that every Borel measurable quasimorphism is at boundeddistance from a quasi-biharmonic one. This allows us to deduce non-degeneratecentral limit theorems and laws of the iterated logarithm for suchquasimorphisms along regular random walks on topological groups using classicalmartingale limit theorems of Billingsley and Stout. For quasi-biharmonicquasimorphism on countable groups we also obtain integral representations usingmartingale convergence.

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