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Spectral theory and limit theorems for geometrically ergodic Markov processes. (English) Zbl 1016.60066
Let $$(\Phi_t)_{t\geq 0}$$ be a time-homogeneous, gometrically ergodic Markov process in continuous or discrete time with a state space $$X$$, and let $$F:X\to \mathbb{R}$$ be a bounded measurable function. The authors investigate the long-time behavior of $$S_t:= \int^t_0 F(\Phi_s)ds$$ in the continuous case and $$S_n:= \sum^n_{i=0} F(\Phi_i)$$ in the discrete case. In particular, the solutions of a so-called multiplicative Poisson equation associated with the transition semigroup of $$(\Phi_t)_{t\geq 0}$$ are studied, and a multiplicative mean ergodic theorem is derived. Moreover, Edgeworth expansions with the leading term of order $$O(t^{-1/2})$$ and large deviation results in a neighborhood of the mean (with exact asymptotic results) are given for $$S_t$$.

##### MSC:
 60J05 Discrete-time Markov processes on general state spaces 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 60F10 Large deviations 60J25 Continuous-time Markov processes on general state spaces
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