Nonconventional limit theorems

I will start with a description of results on various limit theorems for ''nonconventional'' sums of the form $\sum_{n=1}^{N}B(\xi(q_{1}(n)), \xi(q_{2}(n)), \ldots, \xi(q_{\ell} (n)))$, where $\xi(n)$ for some $n \geq 0$ is some stochastic process or a dynamical system while $q_{i}(n)$ are integer valued and are linear for $i \leq k$ and grow faster than linearly for $i > k , i \leq \ell$. The motivation for this study comes, in particular, from many papers about nonconventional ergodic theorems appeared in the last 30 years. Such limit theorems describe multiple recurrence properties of the corresponding processes and dynamical systems and they include central limit theorem, large deviations, averaging which will be discussed in the first part of my talk and Poisson type limit theorems which will form the second part of my talk.

During last 20 years a substantial attention was attracted to the study of numbers of arrivals at small (shrinking) sets by trajectories of dynamical systems during the time intervals inversely proportional to the measure of a set. It seems that the question was originated by Sinai in the framework of the study of distributions of spacings between energy levels of quantum systems. Most of the papers dealt with the symbolic setup of a sequence space $\Omega$ with a shift invariant sufficiently fast mixing probability $P$ where Pitskel, Hirata and Denker showed that when $P$ is a Gibbs shift invariant measure then the numbers of arrivals to shrinking cylindrical neighborhoods of almost all points are asymptotically Poisson distributed. More recently estimates for Poisson approximations were obtained by Abadi and others while Haidn and Vaienti obtained compound Poisson approximations for distributions of numbers of arrivals to shrinking cylindrical neighborhoods of some periodic points. A year and a half ago Poisson limiting behavior in the symbolic situation was extended to numbers of multiple recurrencies (nonconventional setup). In this talk I provide an essentially complete description of possible limiting behaviors of distributions of numbers of multiple recurrencies to shrinking cylinders which is mostly new even for the widely studied single (conventional) recurrencies case (joint work with my student Ariel Rapaport).