### 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) |
---|---|

Pages (from-to) | 649-688 |

Number of pages | 40 |

Journal | Annals of Probability |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - Mar 2014 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annals of Probability*,

*42*(2), 649-688. https://doi.org/10.1214/12-AOP796

**Nonconventional limit theorems in discrete and continuous time via martingales.** / Kifer, Yuri; Varadhan, Srinivasa.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 42, no. 2, pp. 649-688. https://doi.org/10.1214/12-AOP796

}

TY - JOUR

T1 - Nonconventional limit theorems in discrete and continuous time via martingales

AU - Kifer, Yuri

AU - Varadhan, Srinivasa

PY - 2014/3

Y1 - 2014/3

N2 - We obtain functional central limit theorems for both discrete time expressions of the form 1/√NΣ[Nt] n=1(F(X(q1(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 qi(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 qi '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 Xi(n) = Tnfi, 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 Xi(n) = fi(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, Xi(t) = fi(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.

AB - We obtain functional central limit theorems for both discrete time expressions of the form 1/√NΣ[Nt] n=1(F(X(q1(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 qi(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 qi '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 Xi(n) = Tnfi, 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 Xi(n) = fi(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, Xi(t) = fi(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.

KW - Hyperbolic diffeomorphisms

KW - Limit theorems

KW - Markov processes

KW - Martingale approximation

KW - Mixing

UR - http://www.scopus.com/inward/record.url?scp=84894589507&partnerID=8YFLogxK

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U2 - 10.1214/12-AOP796

DO - 10.1214/12-AOP796

M3 - Article

VL - 42

SP - 649

EP - 688

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 2

ER -