Independent linear forms on the group Ω_p

2017/11/24 by Margaryta Myronyuk, Myronyuk, Margaryta
Computer Science · Mathematics · #43A35 #60B15 #62E10 #Advanced Topology and Set Theory #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Topological and Geometric Data Analysis #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.1711.10387

openalex publication_date 2017/11/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \Ω_p be the group of p-adic numbers, \ξ₁, \ξ₂, \ξ₃\nbe independent random variables with values in \Ω_p and distributions\n\μ₁, \μ₂, \μ₃. Let \α_j, \β_j, \γ_j be topological\nautomorphisms of \Ω_p. We consider linear forms L₁ = \α₁\ξ₁ +\n\α₂ \ξ₂+\α₃ \ξ₃, L₂=\β₁\ξ₁ + \β₂ \ξ₂+ \β₃\n\ξ₃ and L₃=\γ₁\ξ₁ + \γ₂ \ξ₂+ \γ₃ \ξ₃. Assuming that\nthe linear forms L₁, L₂ and L₃ are independent, we describe possible\ndistributions \μ₁, \μ₂, \μ₃. This theorem is an analogue of the\nwell-known Skitovich-Darmois theorem, where a Gaussian distribution on the real\nline is characterized by the independence of two linear forms.\n

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