Gerónimo Uribe Bravo

Proof(s) of the Lamperti representation of continuous-state branching processes ∗

This paper uses two new ingredients, namely stochastic differential equations satisfied by continuous-state branching processes (CSBPs), and a topology under which the Lamperti transformation is continuous, in order to provide self-contained proofs of Lampertis 1967 representation of CSBPs in terms of spectrally positive Levy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is evident.

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.

Bridges of Lévy processes conditioned to stay positive

We consider Kallenberg’s hypothesis on the characteristic function of a Levy process and show that it allows the construction of weakly continuous bridges of the Levy process conditioned to stay positive. We therefore provide a notion of normalized excursions Levy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the Levy process conditioned to stay positive by using the weakly continuous bridges of the Levy 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 Levy process, the Durrett–Iglehart theorem stating that the Brownian bridge from 0 to 0 conditioned to remain above −e converges weakly to the Brownian excursion as e → 0, is extended to Levy processes. We also extend the Denisov decomposition of Brownian motion to Levy 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.

Supercritical percolation on large scale-free random trees

We consider Bernoulli bond percolation on a large scale-free tree in the supercritical regime, meaning informally that there exists a giant cluster with high probability. We obtain a weak limit theorem for the sizes of the next largest clusters, extending a recent result in Bertoin [Random Structures Algorithms 44 (2014) 29-44] for large random recursive trees. The approach relies on the analysis of the asymptotic behavior of branching processes subject to rare neutral mutations, which may be of independent interest.

The convex minorant of a Lévy process

We offer a unified approach to the theory of convex minorants of Levy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Levy 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 Levy process with continuous distributions above its convex minorant on the interval [0,1].

LOCAL EXTINCTION IN CONTINUOUS-STATE BRANCHING PROCESSES WITH IMMIGRATION

The purpose of this article is to observe that the zero sets of continuous state branching processes with immigration (CBI) are infinitely divisible regenerative sets. Indeed, they can be constructed by the procedure of random cutouts introduced by Mandelbrot in 1972. We then show how very precise information about the zero sets of CBI can be obtained in terms of the branching and immigrating mechanism.

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.

Convex minorants of random walks and Lévy processes

This article provides an overview of recent work on descriptions and properties of the Convex minorants of random walks and Levy 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 Levy 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.

The comb representation of compact ultrametric spaces

We call a comb a map f: I → [0,∞), where I is a compact interval, such that {f ≥ ε} is finite for any ε > 0. A comb induces a (pseudo)-distance