Integrated Cauchy functional equation and characterizations of the exponential law. (English) Zbl 0584.62019
A general solution of the functional equation \(\int^{\infty}_{0}f(x+y)d\mu (y)=f(x)\) where f is a nonnegative function and \(\mu\) is a \(\sigma\)-finite positive Borel measure on [0,\(\infty)\) is shown to be \(f(x)=p(x)\exp (\lambda x)\) where p is a periodic function with every \(y\in {\underline \mu}\), the support of \(\mu\), as a period.
The solution is applied in characterizing Pareto, exponential and geometric distributions by properties of integrated lack of memory, record values, order statistics and conditional expectation.
The solution is applied in characterizing Pareto, exponential and geometric distributions by properties of integrated lack of memory, record values, order statistics and conditional expectation.
MSC:
62E10 | Characterization and structure theory of statistical distributions |
39B62 | Functional inequalities, including subadditivity, convexity, etc. |
45A05 | Linear integral equations |