NONCONVENTIONAL LIMIT THEOREMS IN DISCRETE AND CONTINUOUS TIME VIA MARTINGALES

• 成果类型: Article
• 署名作者: Kifer, Yuri; Varadhan, S. R. S.
• 署名单位: Hebrew University of Jerusalem; New York University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP796
• 发表日期: 2014
• 页码: 649-688

摘要

We obtain functional central limit theorems for both discrete time expressions of the form 1/root N Sigma([Nt])(n=1) (F(X(q(1)(n)),...,X(q(l)(n))) - (F) over bar) 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, (F) over bar = integral F d(mu X ... X mu), mu 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)(Upsilon(n)), where Upsilon(n) 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)(xi(t)), where xi(t) 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.