Two Novel Characterizations of Self-Decomposability on the Half-Line

Two novel characterizations of self-decomposable Bernstein functions are provided. The first one is purely analytic, stating that a function Ψ is the Bernstein function of a self-decomposable probability law π on the positive half-axis if and only if alternating sums of Ψ satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that Ψ is a self-decomposable Bernstein function if and only if a related d-variate function Cψ,d, ψ:=exp(−Ψ), is a d-variate copula for each d≥2. A canonical stochastic construction is presented, in which π (respectively Ψ) determines the probability law of an exchangeable sequence of random variables {Uk}k∈N such that (U1,…,Ud)∼Cψ,d for each d≥2. The random variables {Uk}k∈N, are i.i.d. conditioned on an increasing Sato process whose law is characterized by Ψ. The probability law of {Uk}k∈N is studied in quite some detail.     «

Two novel characterizations of self-decomposable Bernstein functions are provided. The first one is purely analytic, stating that a function Ψ is the Bernstein function of a self-decomposable probability law π on the positive half-axis if and only if alternating sums of Ψ satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that Ψ is a self-decomposable Bernstein function if and only if a related d-variate function Cψ,d, ψ:=exp(−Ψ), is a d-vari...     »