On analogues of C.R.Rao's theorems for locally compact Abelian groups

TL;DR

This paper extends C.R.Rao's theorem, originally for real-valued variables, to the setting of locally compact Abelian groups and $a$-adic solenoids, characterizing distributions via linear forms.

Contribution

It introduces analogues of C.R.Rao's theorem for independent variables in locally compact Abelian groups and $a$-adic solenoids, with coefficients as continuous endomorphisms.

Findings

Established an analogue of C.R.Rao's theorem for locally compact Abelian groups.

Proved a similar theorem for $a$-adic solenoids.

Demonstrated that the distribution of linear forms determines the original distributions up to a shift.

Abstract

Let $\xi_1$, $\xi_2$, $\xi_3$ be independent random variables with nonvanishing characteristic functions, and $a_j$, $b_j$ be real numbers such that $a_i/b_i\ne a_j/b_j$ for $i\ne j$. Let $L_1=a_1\xi_1+a_2\xi_2+a_3\xi_3$, $L_2=b_1\xi_1+b_2\xi_2+b_3\xi_3$. By C.R.Rao's theorem the distribution of the random vector $(L_1, L_2)$ determines the distributions of the random variables $\xi_j$ up to a change of location. We prove an analogue of this theorem for independent random variables with values in a locally compact Abelian group. We also prove an analogue for independent random variables with values in an $a$-adic solenoid of similar C.R.Rao's theorem. In so doing coefficients of linear forms are continuous endomorphisms of the group.