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 $$\varPsi $$ Ψ is the Bernstein function of a self-decomposable probability law $$\pi $$ π on the positive half-axis if and only if alternating sums of $$\varPsi $$ Ψ satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that $$\varPsi $$ Ψ is a self-decomposable Bernstein function if and only if a related d-variate function $$C_{\psi ,d}$$ C ψ , d , $$\psi :=\exp (-\varPsi )$$ ψ : = exp ( - Ψ ) , is a d-variate copula for each $$d \ge 2$$ d ≥ 2 . A canonical stochastic construction is presented, in which $$\pi $$ π (respectively $$\varPsi $$ Ψ ) determines the probability law of an exchangeable sequence of random variables $$\{U_k\}_{k\in {\mathbb {N}}}$$ { U k } k ∈ N such that $$(U_1,\ldots ,U_d) \sim C_{\psi ,d}$$ ( U 1 , … , U d ) ∼ C ψ , d for each $$d \ge 2$$ d ≥ 2 . The random variables $$\{U_k\}_{k\in {\mathbb {N}}},$$ { U k } k ∈ N , are i.i.d. conditioned on an increasing Sato process whose law is characterized by $$\varPsi $$ Ψ . The probability law of $$\{U_k\}_{k \in {\mathbb {N}}}$$ { U k } k ∈ N is studied in quite some detail.