Independent linear statistics on the cylinders

2012/12/12 by Gennadiy Feldman, Feldman, G. M., Margaryta Myronyuk +1
Computer Science · Mathematics · #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1212.2772

openalex publication_date 2012/12/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let either X=R\timesT or X=Σ_\boldmath a\timesT, where R is the additive group of real number, T is the cycle group and Σ_\boldmath a is an \boldmath a-adic solenoid . Let α_ij, where i, j=1,2,3, be topological automorphisms of the group X. We prove the following analogue of the well-known Skitovich--Darmois theorem for the group X. Let ξ_j, where j=1, 2, 3, be independent random variables with values in the group X and distributions μ_j such that their characteristic functions do not vanish. If the linear statistics L₁=α₁₁ξ₁+α₁₂ξ₂+α₁₃ξ₃, L₂=α₂₁ξ₁+α₂₂ξ₂+α₂₃ξ₃, and L₃=α₃₁ξ₁+α₃₂ξ₂+α₃₃ξ₃ are independent, then all μ_j are Gaussian distributions.

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