On a characterization of the normal distribution by means of identically distributed linear forms (Q1084782)

Let \(X_ 1,X_ 2,...\), be independent, identically distributed random variables. Suppose that the linear forms \(L_ 1=\sum^{\infty}_{j=1}a_ jX_ j\) and \(L_ 2=\sum^{\infty}_{j=1}b_ jX_ j\) exist with probability one and are identically distributed; necessary and sufficient conditions assuring that \(X_ 1\) is normally distributed are presented. The result is an extension of a theorem of \textit{Yu. V. Linnik} [Ukr. Mat. Zh. 5, 207-243, 247-290 (1953; Zbl 0052.367)] concerning the case that the linear forms \(L_ 1\) and \(L_ 2\) have a finite number of nonvanishing components. This proof only makes use of elementary properties of characteristic functions and of meromorphic functions.