## The Annals of Probability

- Ann. Probab.
- Volume 42, Number 2 (2014), 649-688.

### Nonconventional limit theorems in discrete and continuous time via martingales

Yuri Kifer and S. R. S. Varadhan

#### Abstract

We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_{1}(n)),\ldots,X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced by an integral. Here $X(n)$, $n\geq0$ 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, $\bar{F}=\int F\,d(\mu\times\cdots\times\mu)$, $\mu$ is the distribution of $X(0)$ and $q_{i}(n)=in$ for $i\le k\leq\ell$ 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.

#### Article information

**Source**

Ann. Probab., Volume 42, Number 2 (2014), 649-688.

**Dates**

First available in Project Euclid: 24 February 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1393251299

**Digital Object Identifier**

doi:10.1214/12-AOP796

**Mathematical Reviews number (MathSciNet)**

MR3178470

**Zentralblatt MATH identifier**

1304.60041

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60G42: Martingales with discrete parameter 37D99: None of the above, but in this section 60G15: Gaussian processes

**Keywords**

Limit theorems martingale approximation mixing Markov processes hyperbolic diffeomorphisms

#### Citation

Kifer, Yuri; Varadhan, S. R. S. Nonconventional limit theorems in discrete and continuous time via martingales. Ann. Probab. 42 (2014), no. 2, 649--688. doi:10.1214/12-AOP796. https://projecteuclid.org/euclid.aop/1393251299