TY - JOUR
T1 - LOCAL LIMIT THEOREMS FOR PARTIAL SUMS OF STATIONARY SEQUENCES GENERATED BY GIBBS–MARKOV MAPS
AU - AARONSON, JON
AU - DENKER, MANFRED
PY - 2001
Y1 - 2001
N2 - 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.
AB - 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.
U2 - 10.1142/S0219493701000114
DO - 10.1142/S0219493701000114
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SN - 0219-4937
VL - 01
SP - 193
EP - 237
JO - Stochastics and Dynamics
JF - Stochastics and Dynamics
IS - 02
ER -