We show how two change-of-variables formulæ for Lebesgue–Stieltjes integrals generalize when all continuity hypotheses on the integrators are dropped. We find that a sort of “mass splitting phenomenon” arises. Let M : [a, b]→ R be increasing. Then the measure corresponding to M may be defined to be the unique Borel measure μ on [a, b] such that for each continuous function f : [a, b] → R, the i...

By Theorem 2.1 the product φ1φ2 is of bounded variation and by Theorem 2.24 all three Riemann–Stieltjes integrals involved exist as the limit of Riemann–Stieltjes sums as |Γ| → 0. In particular we can find a δ1 > 0 such that ∣∣∣∣∫ b a fφ2 dφ1 −RΓ(fφ2, φ1) ∣∣∣∣ < ε , ∣∣∣∣∫ b a fφ1 dφ2 −RΓ(fφ1, φ2) ∣∣∣∣ < ε for all partitions with |Γ| < δ1. Let Γ = {xi}i=0 be any partition of [a, b] with intermed...

We consider the set of Stieltjes moment sequences, for which every positive power is again a Stieltjes moment sequence, we and prove an integral representation of the logarithm of the moment sequence in analogy to the Lévy-Khintchine representation. We use the result to construct product convolution semigroups with moments of all orders and to calculate their Mellin transforms. As an applicatio...

In this paper we will investigate the semi-markov random processes with positive tendency, negative jumps and delaying boundary at zero in this article. The Laplace transform on time, Laplace-Stieltjes transform on phase of the conditional and unconditional distributions and Laplace-Stieltjes transform of the ergodic distribution are defined. The characteristics of the ergodic distribution will...

This paper presents majorization results for PH-generators. Based on the majorization results, bounds on the moments and Laplace-Stieltjes transforms of phase-type distributions are found. Exponential distributions and Coxian distributions are identified to be extremal PH-distributions with respect to all the moments and Laplace-Stieltjes transforms for certain subsets of PH-distributions. UN C...

This paper presents a formal proof of the Riesz representation theorem in the PVS theorem prover. The Riemann Stieltjes integral was defined in PVS, and the theorem relies on this integral. In order to prove the Riesz representation theorem, it was necessary to prove that continuous functions on a closed interval are Riemann Stieltjes integrable with respect to any function of bounded variation...

The Stieltjes coefficients γk(a) arise in the expansion of the Hurwitz zeta function ζ(s, a) about its single simple pole at s = 1 and are of fundamental and long-standing importance in analytic number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including series representations and summatory relations. Other integral representations provide...

We establish a hierarchy of weighted majorization relations for the singularities of generalized Lamé equations and the zeros of their Van Vleck and Heine-Stieltjes polynomials as well as for multiparameter spectral polynomials of higher Lamé operators. These relations translate into natural dilation and subordination properties in the Choquet order for certain probability measures associated w...