On a characterization of the normal distribution by means of identically distributed linear forms

Let X1, X2,..., be independent, identically distributed random variables. Suppose that the linear forms L1 = [Sigma]j=1[infinity]ajXj and L2 = [Sigma]j=1[infinity]bjXj exist with probability one and are identically distributed; necessary and sufficient conditions assuring that X1 is normally distributed are presented. The result is an extension of a theorem of [4], 207-243, 247-290) concerning the case that the linear forms L1 and L2 have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.