A General Result on Infinite Divisibility

Using and refining a technique developed by O. Thorin, we prove: THEOREM. Let f(x) = C· xβ - 1 h(x), x > 0, be a probability density on (0, ∞). Here β > 0 and h is continuous and satisfies $h(0) = 1$. Assume that h can be analytically continued to the whole complex plane cut along the negative real axis and assume that h satisfies some other regularity assumptions. If h is completely monotone on (0, ∞) and if, for each fixed $u > 0$, the function h(uν(t))h(u/ν(t)), where ν(t) = t + 1 + (t² + 2t)1/2, is completely monotone on (0, ∞), then $f(x)$ is the density of a generalized gamma convolution and hence infinitely divisible. The theorem is applied to show the infinite divisibility of a rather large class of probability densities on (0, ∞). In particular we show that a power with exponent of modulus 1 of the ratio of two gamma distributed rv's has an infinitely divisible distribution.