Mixtures and characteristic functions.

Let g(u|c,y) = exp {ySigmac(k)(u(k) - 1)} with y > 0, Sigma(k=0) (infinity)c(k) = 1,|u| < 1, and c standing for {c(k)}, be a probability generating function of a nonnegative integer-valued random variable. Let S be a distribution function on (0, infinity) non-degenerate at zero. The functions g and S determine another probability generating function, G(u|Sc) = integral(0) (infinity)gdS(y). One of the results obtained asserts that, if the sequence c is finite and the characteristic function of S is entire, then G determines uniquely both S and c. The assertion does not hold if these conditions are not satisfied. Another group of results refers to properties of characteristic functions. Let P(z) be a polynomial of degree m and f(z|y) = exp- {yP(z)}. The theorem of Marcinkiewicz asserts that with m > 2 the function f cannot be a characteristic function. It is shown that, if the characteristic function of S is entire, then F(z) = integral(0) (infinity)f(z|y)dS(y) can be characteristic function only if m </= 2. Again the assertion need not be true if the characteristic function of S is not entire.