Affine relation between an infinitely divisible distribution function and its Lévy measure. (English) Zbl 1465.60018
Summary: A certain subfamily of discrete compound Poisson laws is known to be characterised in terms of an affine relation linking the probability masses with those of its jump law. We show that the continuous version of this relation leads to the so-called Geeta laws and the Kendall-Ressel law (for the zero hitting time of a drifted gamma process). This yields an explicit expression for the Laplace transform of the Kendall-Ressel law in terms of the secondary branch of the Lambert function and it offers an alternative perspective on a recent treatment of the gamma process hitting time. The ambit of logarithmic complete monotonicity of a certain family of functions is extended.
MSC:
| 60E07 | Infinitely divisible distributions; stable distributions |
| 60G51 | Processes with independent increments; Lévy processes |
Keywords:
infinite divisibility; generalized negative-binomial convolution; generalized gamma convolution; spectrally positive Lévy processes; Kendall identity; logarithmically completely monotone functionsSoftware:
DLMFReferences:
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