# Is it true that for every signed probability distribution f, there are positive distributions g and h st. fg=h?

While reading the article Half of a Coin: Negative Probabilities, I came across the following theorem:

For every generalized g.f. f (of a signed probability distribution) there exist two p.d.f.’s g and h (of ordinary nonnegative probability distributions) such that the product fg = h.

This seems like a very useful way to think of signed probability distributions, if it is true. However I weren't able to find any mentions of this theorem outside the article. The author refers to his own paper: Convolution quotients of nonnegative functions. The paper is published, but not freely available and doesn't have many citations.

My question is if you are able to tell whether this theorem is true? Or can help me ascertain whether it's not?

• First of all, I guess that p.d.f. shall be g.f. there. I also think that this link shall work fine. What I do not understand, as $X+Y = Z$ comes as one possible meaning of negative probabilities there, but one does not really need negative probabilities for that. – Ilya Dec 8 '14 at 20:15
• Yes, I don't really understand why Z-Y would have negative probabilities for anything. On the other hand, letting f=h*(-g) surely won't give a solution to fg=h. I guess because of independence? – Thomas Ahle Dec 8 '14 at 23:49