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=dX+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.
Highlights
We deal with probability distributions on the right half-line [0, ∞) and their characterid zation properties expressed in the form of distributional equations of the type Z = X + TZ, where the random variable T ≥ 0 has known distribution, the distribution of the random variable Z ≥ 0 is a transformation of that of X ≥ 0, and we want to find the distribution of
Our goal is to provide necessary and sufficient conditions for such a functional equation to have a unique solution
The unique solution is equivalent to a characterization property of a probability distribution
Summary
We study a wide class of power-mixture functional equations for the LS transforms of probability distributions. If X2 ∼ F2 , where F2 ∈ Exp(1), the standard exponential distribution, F2 (y) = 1 − e−y , y ≥ 0, its LS transform is F2 (s) = 1/(1 + s), s ≥ 0, and the generating distribution F for the composition process ( X (t))t≥0 reduces to the so-called compound-exponential distribution whose LS transform (for short, compound-exponential transform) is This shows that the power-mixture transforms are more general than the compoundexponential ones. Our main purpose in this paper is to provide necessary and sufficient conditions for the functional equations in question to have unique distributional solutions. We do this under quite general conditions, one of them is to require finite variance. The list of references includes significant works all related to our study
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