Nonconventional Poisson limit theorems

The classical Poisson theorem says that if ξ ₁, ξ ₂, ... are i.i.d. 0-1 Bernoulli random variables taking on 1 with probability p _n ≡ λ/n, then the sum S _n = Σ _i=1 ⁿ ξ _i is asymptotically in n Poisson distributed with the parameter λ. It turns out that this result can be extended to sums of the form S_n = ∑_{i = 1}ⁿξ_q1(i) ... ξ_qℓ(i) where now p_n ≡ (lambda;/n)^1/ℓ and 1 ≤ q1⁽ⁱ⁾ < ... < qℓ⁽ⁱ⁾ are integer-valued increasing functions. We obtain also the Poissonian limit for numbers of arrivals to small sets of ℓ-tuples X_q1(i), ... X_qℓ(i) for some Markov chains X _n and for numbers of arrivals of T^q1(i)x, ..., T^qℓ(i)x to small cylinder sets for typical points x of a sub-shift of finite type T.