ar X iv : s ub m it / 03 35 16 7 [ m at h . PR ] 1 0 O ct 2 01 1 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 pn ≡ λ/n then the sum Sn = n i=1 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 Sn = n i=1 ξ q 1 (i) · · · ξ q ℓ (i) where now pn ≡ (λ/n) 1/ℓ and 1 ≤ q 1 (i) < · · · < q ℓ (i) are integer valued increasing functions. We obtain also Poissonian limit for numbers of arrivals to small sets of ℓ-tuples X q 1 (i) , ..., X q ℓ (i) for some Markov chains Xn and for numbers of arrivals of T q 1 (i) x, ..., T q ℓ (i) x to small cylinder sets for typical points x of a subshift of finite type T .