From invariance under binomial thinning to unification of the Cauchy and the Go{\l}\k{a}b-Schinzel-type equations

TL;DR

This paper explores the relationship between invariance properties of probability distributions under binomial thinning and certain functional equations, unifying classical equations like Cauchy and Golab-Schinzel.

Contribution

It establishes a connection between invariance in probability distributions and functional equations, providing solutions under minimal regularity assumptions.

Findings

Connected binomial thinning invariance to functional equations

Unified Cauchy and Golab-Schinzel equations through this framework

Solved these equations with mild regularity conditions

Abstract

We point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Go{\l}ab-Schinzel equation. We solve these equations in several settings with no or mild regularity assumptions imposed on unknown functions.