Summary
LetE be the class ofpdf'sf on (0,∞) such that, for eachu>0,f(uv) f (u/v) is completely monotone as a function ofw=v+v −1. This class includes many familiarpdf's and is closed with respect to multiplication and division of independentrv's. Further,E ⊂T, whereT is the class of generalized Gamma convolutions (GGC) introduced by O. Thorin. Moreover,E coincides with the class ofpdf's of the form\(C \cdot x^{\beta - 1} \cdot \mathop \Pi \limits_{i = 1}^N (1 + c_i x)^{y_i }\) (all parameters positive) or limits thereof. The Laplace transform ϕ of aGGC is characterized by complete monotonicity of ϕ(uv) ϕ(u/v) as a function ofw. This characterization has many consequences and applications. It follows that also the classT has simple multiplicative properties.
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Bondesson, L. Generalized Gamma convolutions and complete monotonicity. Probab. Th. Rel. Fields 85, 181–194 (1990). https://doi.org/10.1007/BF01277981
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DOI: https://doi.org/10.1007/BF01277981
Keywords
- Stochastic Process
- Convolution
- Probability Theory
- Mathematical Biology
- Generalize Gamma