Probability distribution functions of the Grincevičjus series

Given a sequence (ξn,ηn)(ξn,ηn) of independent identically distributed vectors of random variables we consider the Grincevičjus series∑n=1∞ηn∏k=1n−1ξk and a functional-integral equation connected with it. We prove that the equation characterizes all probability distribution functions of the Grincevičjus series. Moreover, some application of this characterization to a continuous refinement equation is presented.