Let P(z) be a polynomial of
degree m > 2 and g(z) an entire function of order less than m. According to a result
of Marcinkiewicz the function g(z)exp{P(z)} cannot be the characteristic function of
a probability distribution. The special case, that exp{P(z)} cannot be a
characteristic function, is generally known as Marcinkiewicz’s theorem. In the present
paper it is shown that if f(z) is any nonconstant entire function then neither
g(z)f[exp{P(z)}] nor f{P(z)} can be characteristic functions. Also, necessary and
sufficient conditions are discussed for functions of the form f[exp{P(z)}] to be
characteristic functions.
