Characterization of distributions by the method of intensively monotone operators (Q801414)

The authors observe that many problems in the field of characterizations (of probability distributions) amount to finding all functions f (in a certain class \(\epsilon\) of continuous functions, that satisfy a functional equation of the form \(Af=f\), where A is an operator. Here f can be a density function, a Laplace transform, or a characteristic function. It is known that \(Af=f\) for all f in a certain subclass \(\{f_{\lambda}\}_{\lambda \in \Lambda}\) of \(\epsilon\), it may often happen that every f that satisfies the functional equation equals \(f_{\lambda}\) for some \(\lambda\). The authors present conditions on A (intensive monotonicity) and conditions on \(\{f_{\lambda}\}\) with respect to \(\epsilon\) (strong \(\epsilon\)-positivity) which guarantee that every solution of the functional equation coincides with \(f_{\lambda}\) for some \(\lambda\). Then they apply the general results to a large number of more or less classical characterization problems.