The Heyde theorem on a group ℝⁿ× D, where D is a discrete Abelian group

2020/07/23 by Myronyuk, Margaryta · 1 citation
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 60B15 #Secondary 62E10 #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2007.12241

Abstract

Heyde proved that a Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear statistic given another. The present article is devoted to a group analogue of the Heyde theorem. We describe distributions of independent random variables ξ₁, ξ₂ with values in a group X=ℝⁿ× D, where D is a discrete Abelian group, which are characterized by the symmetry of the conditional distribution of the linear statistic L₂ = ξ₁ + δξ₂ given L₁ = ξ₁ + ξ₂, where δ is a topological automorphism of X such that Ker(I+δ)=\0\.

Cited by

• Characterization of Probability Distributions on Locally Compact Abelian Groups by the Property of Identical Distribution of Linear Forms with Random Coefficients

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