Probabilistic Cauchy functional equations

Abstract

In this note, we introduce probabilistic Cauchy functional equations, specifically, functional equations of the following form:

$$f(X_1+X_2)\stackrel{\text{d}}{=}f(X_1)+f(X_2),$$

where $X_1$ and $X_2$ represent two independent identically distributed real-valued random variables governed by a distribution μ having appropriate support. The symbol $\stackrel{d}{=}$ denotes equality in distribution. When μ is 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.