Random variables, distributions, functional equations

If X is a random variable with a distribution function F, it is always useful to have properties which characterize uniquely F, hence also X. All started almost a century ago when Polya, Cramer and Raikov established characterizations of the normal and the Poisson distributions. Over the years, characterization problems have been successfully solved by diverse methods. One of them is to use functional equations in terms of Laplace-Stieltjes transforms. In this talk we show that there are classes of functional equations related to nonlinear equations of the type Z = X + TZ, where X, T, Z are nonnegative random variables and “=” means equality in distribution. The problem is to find the distribution function F of X assuming that the distribution of T is known, while the distribution of Z is defined via F. It is important to mention that the uniqueness of the solution of a functional equation (“fixed point”) is equivalent to a characterization property of a distribution. We are going to present new results or extensions of previous results. In particular, we provide another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We give explicit illustrative examples and deal with related topics.
Recording: https://youtu.be/V9MskocZBsU