Selfdecomposability of Weak Variance Generalised Gamma Convolutions

Weak variance generalised gamma convolution processes are multivariate Brownian motions weakly subordinated by multivariate Thorin subordinators. Within this class, we extend a result from strong to weak subordination that a driftless Brownian motion gives rise to a self-decomposable process. Under moment conditions on the underlying Thorin measure, we show that this condition is also necessary. We apply our results to some prominent processes such as the weak variance alpha-gamma process, and illustrate the necessity of our moment conditions in some cases

Disintegration of positive isometric group representations on Lp\mathrm{L}^p-spaces

Let $G$ be a Polish locally compact group acting on a Polish space $X$ with a $G$-invariant probability measure $\mu$. We factorize the integral with respect to $\mu$ in terms of the integrals with respect to the ergodic measures on $X$, and show that $\mathrm{L}^p(X,\mu)$ ($1\leq p<\infty$) is $G$-equivariantly isometrically lattice isomorphic to an $\mathrm{L}^p$-direct integral of the spaces $\mathrm{L}^{p}(X,\lambda)$, where $\lambda$ ranges over the ergodic measures on $X$. This yields a disintegration of the canonical representation of $G$ as isometric lattice automorphisms of $\mathrm{L}^p(X,\mu)$ as an $\mathrm{L}^p$-direct integral of order indecomposable representations. If $(X^\prime,\mu^\prime)$ is a probability space, and, for some $1\leq q<\infty$, $G$ acts in a strongly continuous manner on $\mathrm{L}^q(X^\prime,\mu^\prime)$ as isometric lattice automorphisms that leave the constants fixed, then $G$ acts on $\mathrm{L}^{p}(X^{\prime},\mu^{\prime})$ in a similar fashion for all $1\leq p<\infty$. Moreover, there exists an alternative model in which these representations originate from a continuous action of $G$ on a compact Hausdorff space. If $(X^\prime,\mu^\prime)$ is separable, the representation of $G$ on $\mathrm{L}^p(X^\prime,\mu^\prime)$ can then be disintegrated into order indecomposable representations. The notions of $\mathrm{L}^p$-direct integrals of Banach spaces and representations that are developed extend those in the literature.Comment: Section on future perspectives added. 35 pages. To appear in Positivit

On the Limit Distributions of Continuous-State Branching Processes with Immigration

We consider the class of continuous-state branching processes with immigration (CBI-processes), introduced by Kawazu and Watanabe (1971) and their limit distributions as time tends to infinity. We determine the Levy-Khintchine triplet of the limit distribution and give an explicit description in terms of the characteristic triplets of the Levy subordinator and the spectrally positive Levy process, which describe the immigration resp. branching mechanism of the CBI-process. This representation allows us to describe the support of the limit distribution and characterise its absolute continuity and asymptotic behavior at the boundary of the support, generalizing several known results on self-decomposable distributions.Comment: minor update; to appear in SP