## Abstract

We obtain functional central limit theorems for both discrete time expressions of the form 1/√NΣ^{[Nt]}_{n=1}(F(X(q_{1}(n)),., X(qℓ(n)))-F̄) and similar expressions in the continuous time where the sum is replaced by an integral. Here X(n),n≥0 is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, F is a continuous function with polynomial growth and certain regularity properties, F̄=∫F d(μ×.×μ), μ is the distribution of X(0) and q_{i}(n) = in for i≤k≤ℓ while for i > k they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when q_{i} 's are polynomials of increasing degrees. These results decisively generalize [Probab. Theory Related Fields 148 (2010) 71-106], whose method was only applicable to the case k = 2 under substantially more restrictive moment and mixing conditions and which could not be extended to convergence of processes and to the corresponding continuous time case. As in [Probab. Theory Related Fields 148 (2010) 71-106], our results hold true when X_{i}(n) = T^{n}f_{i}, where T is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when X_{i}(n) = f_{i}(γn), where γn is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure. Moreover, our relaxed mixing conditions yield applications to other types of dynamical systems and Markov processes, for instance, where a spectral gap can be established. The continuous time version holds true when, for instance, X_{i}(t) = f_{i}(ξt), where ξt is a nondegenerate continuous time Markov chain with a finite state space or a nondegenerate diffusion on a compact manifold. A partial motivation for such limit theorems is due to a series of papers dealing with nonconventional ergodic averages.

Original language | English (US) |
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Pages (from-to) | 649-688 |

Number of pages | 40 |

Journal | Annals of Probability |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2014 |

## Keywords

- Hyperbolic diffeomorphisms
- Limit theorems
- Markov processes
- Martingale approximation
- Mixing

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty