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Characterization of distributions by the method of intensively monotone operators. (English) Zbl 0551.62008
Lecture Notes in Mathematics. 1088. Berlin etc.: Springer-Verlag. X, 175 p. DM 26,50 (1984).
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.
Reviewer: L.Bondesson

MSC:
62E10 Characterization and structure theory of statistical distributions
62-02 Research exposition (monographs, survey articles) pertaining to statistics
47H05 Monotone operators and generalizations
62H05 Characterization and structure theory for multivariate probability distributions; copulas
47H10 Fixed-point theorems
47J05 Equations involving nonlinear operators (general)
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