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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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