We discuss various limit theorems for "nonconventional" sums of the form $\sum ^N_{n=1}F\left ( \xi \left ( n \right ),\xi \left ( 2n \right ),...,\xi \left ( \ell n \right ) \right )$ where $\xi \left ( n \right )$ is a stochastic process or a dynamical system. 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 corresponding stochastic processes and dynamical systems. Among our results are: central limit theorem, a.s. central limit theorem, local limit theorem, large deviations and averaging. Some multifractal type questions and open problems will be discussed, as well.
Keywords : limit theorems - nonconventional sums - multiple recurrence[-]

The Poisson limit theorem which appeared in 1837 seems to be the first law of rare events in probability. Various generalizations of it and estimates of errors of Poisson approximations were obtained in probability and more recently this became a popular topic in dynamics in the form of study of asymptotics of numbers of arrivals at small (shrinking) sets by a stochastic process or by a dynamical system. I will describe recent results on Poisson and compound Poisson asymptotics in a nonconventional setup, i.e. for numbers of events of multiple returns to shrinking sets, namely, for numbers of combined events of the type $\left \{ \omega : \xi \left ( jn,\omega\right )\in \Gamma_N,j = 1,...,\ell \right \},n\leq N$ where $\xi \left ( k,\omega \right )$ is defined as a stochastic process from the beginning or it is built from a dynamical system by writing $\xi \left ( k,\omega \right )=T^k\omega .$ We obtain an essentially complete description of possible limiting behaviors of distributions of numbers of multiple recurrencies to shrinking cylinders for $\psi $-mixing shifts. Some possible extensions and related questions will be discussed, as well. Most of the results were obtained jointly with my student Ariel Rapaport and some of them are new even for the widely studied single (conventional) recurrencies case.
Keywords : Poisson limit theorems - nonconventional sums - multiple recurrence[-]