Limit theorems for numbers of multiple returns in nonconventional arrays

2019/10/03 by Yuri Kifer, Kifer, Yuri
Computer Science · Mathematics · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1910.01439

openalex publication_date 2019/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a ψ-mixing process ξ₀,ξ₁,ξ₂,... we consider the number N_N of multiple returns \ξ_q_i,N(n)∈Γ_N, i=1,...,ℓ\ to a set Γ_N for n until either a fixed number N or until the moment τ_N when another multiple return \ξ_q_i,N(n)∈Δ_N, i=1,...,ℓ\ takes place for the first time where Γ_N∩Δ_N=∅ and q_i,N, i=1,...,ℓ are certain functions of n taking on nonnegative integer values when n runs from 0 to N. The dependence of q_i,N(n)'s on both n and N is the main novelty of the paper. Under some restrictions on the functions q_i,N we obtain Poisson distributions limits of N_N when counting is until N as N→∞ and geometric distributions limits when counting is until τ_N as N→∞. We obtain also similar results in the dynamical systems setup considering a ψ-mixing shift T on a sequence space Ω and studying the number of multiple returns \ T^q_i,N(n)ω∈ A^a_n, i=1,...,ℓ\ until the first occurrence of another multiple return \ T^q_i,N(n)ω∈ A^b_m, i=1,...,ℓ\ where A^a_n, A_m^b are cylinder sets of length n and m constructed by sequences a,b∈Ω, respectively, and chosen so that their probabilities have the same order.

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