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

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.