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

Chin-Yuan Hu, Gwo Dong Lin and Jordan M. Stoyanov
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Chin-Yuan Hu: National Changhua University of Education, Changhua 50058, Taiwan
Gwo Dong Lin: Social and Data Science Research Center, Hwa-Kang Xing-Ye Foundation, Taipei 10659, Taiwan
Jordan M. Stoyanov: Institute of Mathematics & Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

Mathematics, 2021, vol. 9, issue 3, 1-21

Abstract: We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line [ 0 , ? ) . These equations arise when studying distributional equations of the type Z = d X + T Z , where the random variable T ? 0 has known distribution, while the distribution of the random variable Z ? 0 is a transformation of that of X ? 0 , and we want to find the distribution of 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 that 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.

Keywords: distributional equation; Laplace–Stieltjes transform; Bernstein function; power-mixture transform; functional equation; characterization of distributions (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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