Some general random Taylor series

Abstract

Kahane has studied the value distribution of Gauss-Taylor series Σ a n X n z n where { X n }is a complex Gauss sequence and Σ a 2 n =∞The value distribution of more general random Taylor series is considered, where { X n } is a sequence of real or complex random variables of independent, symmetric and equally distributed with finite non-zero fourth moment (the classical Gauss, Steinhaus and Rademacher random variables are special cases of such variables). First a theorem on the growth of characteristic functions is proved by a method which is completely different from Kahane's. Then it is applied to proving that the range of general random Taylor series is almost surely dense everywhere in the complex plane and that if the random variable is bounded and continuous, the random series surely has no finite Nevanlinna deficient value.