On a characterisation theorem for probability distributions on discrete Abelian groups

2018/04/11 by Feldman, G. M.
#43A35 #60B15 #62E10 #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.1804.04508

Abstract

Let X be a countable discrete Abelian group containing no elements of order 2, α be an automorphism of X, ξ₁ and ξ₂ be independent random variables with values in the group X and distributions μ₁ and μ₂. The main result of the article is the following statement. The symmetry of the conditional distribution of the linear form L₂ = ξ₁ + αξ₂ given L₁ = ξ₁ + ξ₂ implies that μ_j are shifts of the Haar distribution of a finite subgroup of X if and only if the automorphism α satisfies the condition \rm Ker(I+α)=\0\. This theorem is an analogue for discrete Abelian groups the well-known Heyde theorem where Gaussian distribution on the real line is characterized by the symmetry of the conditional distribution of one linear form of independent random variables given another. We also prove some generalisations of this theorem.

Related

• Characterization of probability distributions on some locally compact Abelian groups containing an element of order 2
• The Heyde theorem on a group ℝⁿ× D, where D is a discrete Abelian group
• On a characterization of shifts of Haar distributions on compact open subgroups of a compact Abelian group
• Generalization of the Heyde theorem to finite Abelian groups and groups of the form RxG, where G is a finite Abelian group
• The Heyde characterization theorem on some locally compact Abelian groups