On C.R. Rao’s Theorem for Locally Compact Abelian Groups

Abstract

Let ξ ₁, ξ₂, ξ₃ be independent random variables with values in a locally compact Abelian group X with nonvanishing characteristic functions, and a_j, b_j be continuous endomorphisms of X satisfying some restrictions. Let L₁ = a₁ξ ₁+ a₂ ξ₂ + a₃ξ ₃, L₂ = b₁ξ ₁ + b₂ξ ₂ + b₃ξ ₃. It was proved that the distribution of the random vector (L₁, L₂ ) determines the distributions of the random variables ξ_j up a shift. This result is a group analogue of the well-known C.R. Rao theorem. We also prove an analogue of another C.R. Rao's theorem for independent random variables with values in an a-adic solenoid.