Integrated Cauchy functional equation and characterizations of the exponential law

Abstract

A general solution of the functional equation ∫∞(x+y)dμ(y)=f(x) where f is a nonnegative function and μ is a σ-finite positive Borel measure on [0,∞) is shown to be f(x)exp(λ x) where p is a periodic function with every y∈μ, the support of μ as a period. The solution is applied in characterrizing Pareto, exponential and geometric distributions by properties of integrated lack of memory, record values, order statistics and conditional expectation.