Bernstein Functions at Work: Coalescents, Copulas, and Subordination

Several positivity questions in stochastic processes, dependence modeling, fractional analysis, and renewal theory reduce to a common recognition task: after normalization, identify the object as a Laplace transform, a potential density, an inverse-flow coefficient, or a finite kernel average, and then read the sign pattern from that representation. We develop this recognition calculus for completely monotone functions, Bernstein functions, special Bernstein functions, and probabilistic realizations through subordinators and mixing measures. The main affirmative results settle three narrowly stated source questions in the conventions used by their source papers. M\"ohle's Problem 6.3 on the block-counting process of exchangeable coalescents with residual singleton mass (dust) is proved by a finite-simplex ordered-pair kernel certificate. For the Pearse--Bondell power-divergence copula generators, we prove complete monotonicity of the inverse throughout the remaining strict negative range $\lambda\le-1$ identified in their Section 3.8. Together with the special cases already verified in the source paper, this yields Archimedean copulas in every dimension for $\lambda\le-1$. The Bendikov--Cygan monotonicity question for discrete renewal sequences attached to special Bernstein functions is answered by representing the potential kernel as a Gamma average of a nonincreasing density. Supporting representation and boundary results cover Sibisi's Prabhakar--Pollard $Q$-measure, the Mecke--Nagel--Weiss atom at zero, and the cubic branch criterion $a^2\ge3b$.