Distributional Equation Laplace–Stieltjes Transform Bernstein Function Power-mixture Transform Functional Equation Characterization of Distributions
Issue Date:
29-Jan-2021
Publisher:
MDPI
Citation:
Hu, C.-Y.; Lin, G.D.; Stoyanov, J.M. Characterization of Probability Distributions via Functional Equations of Power-Mixture Type. Mathematics, 2021, 9, 271. https://doi.org/10.3390/math9030271
Series/Report no.:
Mathematics;9, 271
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 \( Z \overset{d}{=} X + TZ \), 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.
