Nonconventional limit theorems in discrete and continuous time via martingales

Yuri Kifer, S. R.S. Varadhan

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish (US)
Pages (from-to)649-688
Number of pages40
JournalAnnals of Probability
Volume42
Issue number2
DOIs
StatePublished - Mar 2014

Keywords

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

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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