Abstract
SupposeL(X) is the law of a positive random variableX, andZ is positive and independent ofX. Admissible solution pairs (L(X),L(Z)) are sought for the in-law equation\(\hat X \cong X o Z\) ∘Z, where\(L\left( {\hat X} \right)\) is a weighted law constructed fromL(X), and ∘ is a binary operation which in some sense is increasing. The class of weights includes length biasing of arbitrary order. When ∘ is addition and the weighting is ordinary length biasing, the class of admissibleL(X) comprises the positive infinitely divisible laws. Examples are given subsuming all known specific cases. Some extensions for general order of length-biasing are discussed.
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Pakes, A.G., Sapatinas, T. & Fosam, E.B. Characterizations, length-biasing, and infinite divisibility. Statistical Papers 37, 53–69 (1996). https://doi.org/10.1007/BF02926159
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DOI: https://doi.org/10.1007/BF02926159