Nonconventional Poisson Limit Theorems

The classical Poisson theorem says that if $\xi_1,\xi_2,...$ are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability $p_n\equiv \la/n$ then the sum $S_n=\sum_{i=1}^n\xi_i$ is asymptotically in $n$ Poisson distributed with the parameter $\la$. It turns out that this result can be extended to sums of the form $S_n=\sum_{i=1}^n\xi_{q_1(i)}... \xi_{q_\ell(i)}$ where now $p_n\equiv(\la/n)^{1/\ell}$ and $1\leq q_1(i) <... <q_\ell(i)$ are integer valued increasing functions. We obtain also Poissonian limit for numbers of arrivals to small sets of $\ell$-tuples $X_{q_1(i)},...,X_{q_\ell(i)}$ for some Markov chains $X_n$ and for numbers of arrivals of $T^{q_1(i)}x,...,T^{q_\ell(i)}x$ to small cylinder sets for typical points $x$ of a subshift of finite type $T$.