On a Pitman-Yor problem

We answer the question posed by Pitman and Yor (Canad. J. Math. 55 (2003) 292-330) about the existence of probability distributions which solve distributional equation of a special type. In the "typical" case we study the tail behaviour of these distributions. Two new explicit examples are given.Perpetuity Fixed points Shot noise transform Shot noise distribution Infinite divisibility Tail behaviour