On C.R. RAO’s theorem for locally compact abelian groups
Let x1, x2, x3 be independent random variables with values in a locally compact Abelian group X with nonvanishing characteristic functions, and aj, bj be continuous endomorphisms of X satisfying some restrictions. Let L1 = a1x1 + a2x2 + a3x3, L2 = b1x1 + b2x2 + b3x3. It was proved that the distribution of the random vector (L1; L2) determines the distributions of the random variables xj 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.