Structural properties of generalised Planck distributions

A family of generalised Planck (GP) laws is defined and its structural properties explored. Sometimes subject to parameter restrictions, a GP law is a randomly scaled gamma law; it arises as the equilibrium law of a perturbed version of the Feller mean reverting diffusion; the density functions can be decreasing, unimodal or bimodal; it is infinitely divisible. It is argued that the GP law is not a generalised gamma convolution. Characterisations are obtained in terms of invariance under random contraction of a weighted version of a related law. The GP law is a particular instance of equilibrium laws obtained from a recursion suggested by a genetic mutation-selection balance model. Some related infinitely divisible laws are exhibited.

Approximation of free convolutions by free infinitely divisible laws

Основываясь на методе сyбoрдинационных функций, мы получаем оценки минимальных ошибок аппроксимации $n$-кратных свободных сверток вероятностных мер безгранично делимыми свободными вероятностными мерами.

A Note on the Gaps in the Support of Discretely Infinitely Divisible Laws

We discuss the nature of gaps in the support of a discretely infinitely divisible distribution from the angle of compound Poisson laws/processes. The discussion is extended to infinitely divisible distributions on the nonnegative real line.