2011/10/10 by Yuri Kifer, Kifer, Yuri
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #math.PR #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1110.2155
14 pages
arxiv created 2011/10/10 · openalex publication_date 2011/10/10 · arxiv updated 2011/10/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical Poisson theorem says that if ξ1,ξ2,... are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability pn≡ \la/n then the sum Sn=∑i=1nξi is asymptotically in n Poisson distributed with the parameter \la. It turns out that this result can be extended to sums of the form Sn=∑i=1nξq1(i)... ξq_ℓ(i) where now pn≡(\la/n)1/ℓ and 1≤ q1(i) <... <q_ℓ(i) are integer valued increasing functions. We obtain also Poissonian limit for numbers of arrivals to small sets of ℓ-tuples Xq1(i),...,Xq_ℓ(i) for some Markov chains Xn and for numbers of arrivals of Tq1(i)x,...,Tq_ℓ(i)x to small cylinder sets for typical points x of a subshift of finite type T.