On C.R. RAO’s theorem for locally compact abelian groups
Let x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub> be independent random variables with values in a locally compact Abelian group <i>X </i>with nonvanishing characteristic functions, and <i>aj</i>, <i>bj </i>be continuous endomorphisms of <i>X </i>satisfying some restrictions. Let <i>L</i>1 = <i>a</i>1x<sub>1</sub> + <i>a</i>2x<sub>2</sub> + <i>a</i>3x<sub>3</sub>, <i>L</i>2 = <i>b</i>1x<sub>1</sub> + <i>b</i>2x<sub>2</sub> + <i>b</i>3x<sub>3</sub>. It was proved that the distribution of the random vector (<i>L</i>1; <i>L</i>2) determines the distributions of the random variables x<i><sub>j</sub></i> 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 <i>a</i>-adic solenoid.