Vol. 20, No. 2, 1967

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Generalization of a theorem of Marcinkiewicz

H. D. Miller

Vol. 20 (1967), No. 2, 261–274

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.

Mathematical Subject Classification
Primary: 60.20
Received: 16 March 1965
Published: 1 February 1967
H. D. Miller