TY - JOUR
T1 - Nonconventional limit theorems in discrete and continuous time via martingales
AU - Kifer, Yuri
AU - Varadhan, S. R.S.
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
U2 - 10.1214/12-AOP796
DO - 10.1214/12-AOP796
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AN - SCOPUS:84894589507
SN - 0091-1798
VL - 42
SP - 649
EP - 688
JO - Annals of Probability
JF - Annals of Probability
IS - 2
ER -