LOCAL LIMIT THEOREMS FOR PARTIAL SUMS OF STATIONARY SEQUENCES GENERATED BY GIBBS–MARKOV MAPS

We introduce Gibbs–Markov maps T as maps with a (possibly countable) Markov partition and a certain type of bounded distortion property, and investigate its Frobenius–Perron operator P acting on (locally) Lipschitz continuous functions ϕ. If such a function ϕ belongs to the domain of attraction of a stable law of order in (0,2), we derive the expansion of the eigenvalue function t↦λ(t) of the characteristic function operators Ptf=Pfexp[i< t,ϕ> (perturbations of P) around 0. From this representation local and distributional limit theorems for partial sums ϕ+…+ϕ◦ Tn are easily obtained, provided ϕ is aperiodic. Applications to recurrence properties of group extensions are also given.