arXiv subjects

Gerónimo Uribe Bravo

Publications and source records attributed to Gerónimo Uribe Bravo.

24 records · Page 2Linked to original sources

arXiv2014

Bridges of Lévy processes conditioned to stay positive

We consider Kallenberg's hypothesis on the characteristic function of a Lévy process and show that it allows the construction of weakly continuous bridges of the Lévy process conditioned to stay positive. We therefore provide a notion of normalized excursions Lévy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Lévy process conditioned to stay positive by using the weakly continuous bridges of the Lévy process itself. For this, we rely on a method due to Hunt which had only been shown to provide upper semi-continuous versions. Using the bridges of the conditioned Lévy process, the Durrett-Iglehart theorem stating that the Brownian bridge from $0$ to $0$ conditioned to remain above $-\varepsilon$ converges weakly to the Brownian excursion as $\varepsilon \to0$, is extended to Lévy processes. We also extend the Denisov decomposition of Brownian motion to Lévy processes and their bridges, as well as Vervaat's classical result stating the equivalence in law of the Vervaat transform of a Brownian bridge and the normalized Brownian excursion.

math.PR↗

arXiv2013

A Lamperti-type representation of continuous-state branching processes with immigration

Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Levy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman's limit theorem concerning Galton-Watson processes conditioned on total population size to more general offspring laws.

math.PR↗

arXiv2012

The convex minorant of a Lévy process

We offer a unified approach to the theory of convex minorants of Lévy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Lévy process on both finite and infinite time intervals, and of a Poisson point process of excursions above the convex minorant up to an independent exponential time. The Poisson-Dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a Lévy process with continuous distributions above its convex minorant on the interval $[0,1]$.

math.PR↗

arXiv2011

Markovian bridges: Weak continuity and pathwise constructions

A Markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a Markov process given its terminal value. As such, Markovian bridges admit a natural parameterization in terms of the state space of the process. In the context of Feller processes with continuous transition densities, we construct by weak convergence considerations the only versions of Markovian bridges which are weakly continuous with respect to their parameter. We use this weakly continuous construction to provide an extension of the strong Markov property in which the flow of time is reversed. In the context of self-similar Feller process, the last result is shown to be useful in the construction of Markovian bridges out of the trajectories of the original process.

math.PR↗

arXiv2011

Convex minorants of random walks and Lévy processes

This article provides an overview of recent work on descriptions and properties of the convex minorant of random walks and Lévy processes which summarize and extend the literature on these subjects. The results surveyed include point process descriptions of the convex minorant of random walks and Lévy processes on a fixed finite interval, up to an independent exponential time, and in the infinite horizon case. These descriptions follow from the invariance of these processes under an adequate path transformation. In the case of Brownian motion, we note how further special properties of this process, including time-inversion, imply a sequential description for the convex minorant of the Brownian meander.

math.PR↗

arXiv2008

The falling appart of the tagged fragment and the asymptotic disintegration of the Brownian height fragmentation

We present a further analysis of the fragmentation at heights of the normalized Brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the Brownian fragmentation when it is about to reduce to dust are described in a limit theorem.

math.PR↗