Nonconventional Poisson limit theorems

Abstract

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 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 ${S_n} = \sum\nolimits_{i = 1}^n {{\xi _{{q_1}(i)}} \cdots {\xi _{{q_\ell }(i)}}}$ where now ${X_{{q_1}(i), \ldots ,}}{X_{{q_\ell }(i)}}$ and ${T^{{q_1}(i)}}x, \ldots ,{T^{{q_\ell }(i)}}x$ are integer-valued increasing functions. We obtain also the Poissonian limit for numbers of arrivals to small sets of ℓ-tuples ${X_{{q_1}(i), \ldots ,}}{X_{{q_\ell }(i)}}$ for some Markov chains X _n and for numbers of arrivals of ${T^{{q_1}(i)}}x, \ldots ,{T^{{q_\ell }(i)}}x$ to small cylinder sets for typical points x of a sub-shift of finite type T.