New Representation Theorems for Completely Monotone and Bernstein Functions with Convexity Properties on Their Measures

Background Risk Models and Stepwise Portfolio Construction

“…When choosing C or, alternatively, π for portfolio modeling purposes, we may wish, or need, to impose certain shape constraints on them. We note, however, that shape relationships between C and π can be quite complex, as seen from the recent works of Sendov and Zitikis (2014), and Sendov and Shan (2015). Our next BRM follows.…”

Poisson processes and a log-concave Bernstein theorem

“…If f (x) < ∞ for all x ∈ R, then µ is absolutely-continuous with a log-concave density f , as required. (15) implies that f equals +∞ in an interval of positive length, in contradiction to the integrability of f . Thus f (x) = 0 for all x = x 0 , and µ is supported at the point {x 0 }.…”

“…Otherwise, there exists x 0 ∈ R with f (x 0 ) = +∞. Since f (x 0 ) = +∞, necessarily f (x) = 0 for x = x 0 , as otherwise (15) implies that f equals +∞ in an interval of positive length, in contradiction to the integrability of f . Thus f (x) = 0 for all x = x 0 , and µ is supported at the point {x 0 }.…”

“…Hirsch's theorem states that this happens if and only if ϕ takes the form (2) for a measure µ whose density is non-increasing and log-convex, apart from an atom at the origin. See Hirsch [10] and Schilling, Song and Vondraček [14,Section 11.2] for a precise formulation and a proof of Hirsch's theorem, and also Forst [9] and Sendov and Shan [15] for related results.…”

Dirichlet polynomials and a moment problem

“…Hence, by Lemma 5.1, the representing measure of f is given by ϕ * µ (see(3.3)). We now apply[17, Theorem 7.2] to f and the Bernstein function g(x) = log(x + 1), x ∈ [0, ∞), to conclude that the representing measure η of the completely monotone function f • g is given by…”