From invariance under binomial thinning to unification of the Cauchy and the Gołąb-Schinzel-type equations

2021/06/16 by Karol Baron, Baron, Karol, Jacek Wesołowski +1
Computer Science · Mathematics · #39B22 #39B52 #62E10 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical functions and polynomials #Polynomial and algebraic computation #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2106.08665

openalex publication_date 2021/06/16 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

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 Gołab-Schinzel equation. We solve these equations in several settings with no or mild regularity assumptions imposed on unknown functions.

Related

• Cauchy's functional equation and extensions: Goldie's equation and inequality, the Gołąb-Schinzel equation and Beurling's equation
• Study of power series distributions with specified covariances
• Clebsch-Gordan coefficients and the binomial distribution
• A direct approach to the stable distributions
• Distributions with dynamic test functions and multiplication by discontinuous functions