Several authors studied identically distributed linear forms in independently and
identically distributed random variables. J. Marcinkiewicz considered finite or infinite
linear forms and assumed that the random variables have finite moments of all
orders. He showed that the common distribution of the random variables is then the
Normal distribution. Yu. V. Linnik obtained some deep results concerning identically
distributed linear forms involving only a finite number of random variables. The
authors have investigated in a separate paper the case where one of the linear forms
contains infinitely many terms while the other is a monomial. They obtained a
characterization of the normal distribution under the assumption that the
second moment of the random variable is finite. In the present paper we
investigate a similar problem and do not assume the existence of the second
moment.
