Probabilistic Cauchy Functional Equations

TL;DR

This paper introduces probabilistic Cauchy functional equations involving sums of random variables and establishes conditions under which solutions are linear, connecting to classical functional equations in a probabilistic setting.

Contribution

It defines probabilistic Cauchy functional equations, provides conditions for linear solutions when the distribution is exponential, and links to integrated Cauchy equations in the general case.

Findings

Unique linear solutions under exponential distribution with regularity conditions

Connection established between probabilistic and classical Cauchy functional equations

Partial results for general distributions

Abstract

In this short note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form: $$ f(X_1 + X_2) \stackrel{d}{=} f(X_1) + f(X_2), $$ where $X_1$ and $X_2$ represent two independent identically distributed real-valued random variables governed by a distribution $\mu$ having appropriate support on the real line. The symbol $\stackrel{d}{=}$ denotes equality in distribution. When $\mu$ follows an exponential distribution, we provide sufficient (regularity) conditions on the function $f$ to ensure that the unique measurable solution to the above equation is solely linear. Furthermore, we present some partial results in the general case, establishing a connection to integrated Cauchy functional equations.