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arXiv · 2009.09148

Characterization of Probability Distributions via Functional Equations of Power-Mixture Type

Chin-Yuan Hu·Gwo Dong Lin·Jordan M. Stoyanov
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Abstract

We study power-mixture type functional equations in terms of Laplace-Stieltjes transforms of probability distributions. These equations arise when studying distributional equations of the type Z = X + TZ, where T is a known random variable, while the variable Z is defined via X, and we want to `find' X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results which are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.

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Chin-Yuan Hu, Gwo Dong Lin, Jordan M. Stoyanov. 2020-09-19. Characterization of Probability Distributions via Functional Equations of Power-Mixture Type. https://arxiv.org/abs/2009.09148

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