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

• Main Authors: Chin-Yuan Hu, Gwo Dong Lin, Jordan M. Stoyanov
• Format: Article
• Language: English
• Published: MDPI AG 2021-01-01
• Series: Mathematics
• Subjects: distributional equation; Laplace–Stieltjes transform; Bernstein function; power-mixture transform; functional equation; characterization of distributions
• Online Access: https://www.mdpi.com/2227-7390/9/3/271

We study power-mixture type functional equations in terms of Laplace–Stieltjes transforms of probability distributions on the right half-line <inline-formula><math display="inline"><semantics><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> These equations arise when studying distributional equations of the type <inline-formula><math display="inline"><semantics><mrow><mi>Z</mi><mover><mo>=</mo><mi mathvariant="normal">d</mi></mover><mi>X</mi><mo>+</mo><mi>T</mi><mi>Z</mi></mrow></semantics></math></inline-formula>, where the random variable <inline-formula><math display="inline"><semantics><mrow><mi>T</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> has known distribution, while the distribution of the random variable <inline-formula><math display="inline"><semantics><mrow><mi>Z</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> is a transformation of that of <inline-formula><math display="inline"><semantics><mrow><mi>X</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, and we want to find the distribution of <i>X</i>. 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.