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.Comment: 23 pages, 4 figures, AMSLaTeX (PDFLaTeX), accepted in Annales de l'Institut Henri Poincar\'e (B

Bridges of L\'{e}vy processes conditioned to stay positive

We consider Kallenberg's hypothesis on the characteristic function of a L\'{e}vy process and show that it allows the construction of weakly continuous bridges of the L\'{e}vy process conditioned to stay positive. We therefore provide a notion of normalized excursions L\'{e}vy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the L\'{e}vy process conditioned to stay positive by using the weakly continuous bridges of the L\'{e}vy 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 L\'{e}vy 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 L\'{e}vy processes. We also extend the Denisov decomposition of Brownian motion to L\'{e}vy 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.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ481 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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.Comment: 23 pages, 4 figures, AMSLaTeX (PDFLaTeX), accepted in Annales de l'Institut Henri Poincar\'e (B

Shifting processes with cyclically exchangeable increments at random

We propose a path transformation which applied to a cyclically exchangeable increment process conditions its minimum to belong to a given interval. This path transformation is then applied to processes with start and end at zero. It is seen that, under simple conditions, the weak limit as epsilon tends to zero of the process conditioned on remaining above minus epsilon exists and has the law of the Vervaat transformation of the process. We examine the consequences of this path transformation on processes with exchangeable increments, L\'evy bridges, and the Brownian bridge.Comment: 14 pages and 3 figure

The convex minorant of a L\'{e}vy process

We offer a unified approach to the theory of convex minorants of L\'{e}vy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a L\'{e}vy 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 L\'{e}vy process with continuous distributions above its convex minorant on the interval $[0,1]$.Comment: Published in at http://dx.doi.org/10.1214/11-AOP658 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Projections of spherical Brownian motion

We obtain a stochastic differential equation (SDE) satisfied by the first $n$ coordinates of a Brownian motion on the unit sphere in $\mathbb{R}^{n+\ell}$. The SDE has non-Lipschitz coefficients but we are able to provide an analysis of existence and pathwise uniqueness and show that they always hold. The square of the radial component is a Wright-Fisher diffusion with mutation and it features in a skew-product decomposition of the projected spherical Brownian motion. A more general SDE on the unit ball in $\mathbb{R}^{n+\ell}$ allows us to geometrically realize the Wright-Fisher diffusion with general non-negative parameters as the radial component of its solution.Comment: 13 page

Flora cuyana

Fil: Sosa, Gerónimo. Universidad Nacional de Cuyo. Facultad de Filosofía y Letra