On a problem of Kahane about the image of Gaussian Taylor series (Q1092514)

An answer is given to a question posed by \textit{J. P. Kahane} [Some random series of functions (1968; Zbl 0192.538)] about Gaussian Taylor series, i.e. power series of the form \(F(z)=\sum^{\infty}_{n=0}a_ nZ_ nz^ n\) where \(a_ n\) is a series of positive constants satisfying certain conditions and \(\{Z_ n\}\) is a sequence of independent complex normals in standard form. It is shown that if \(\sum a^ 2_ n\) diverges then F(z) takes a.s. every complex value with at most one exception. An extension is given, some conjectures noted and the differences between Gaussian and classical Taylor series illustrated by an example.