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Sep 22, 2013
09/13

by
Gerónimo Uribe Bravo

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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...

Source: http://arxiv.org/abs/0811.4754v1

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40

Sep 22, 2013
09/13

by
Gerónimo Uribe Bravo

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We consider Kallenberg's hypothesis on the characteristic function of a L\'evy process and show that it allows the construction of weakly continuous bridges of the L\'evy process conditioned to stay positive. We therefore provide a notion of normalized excursions of L\'evy processes above their cumulative minimum. Our main contribution is the construction of a continuous version of the transition density of the L\'evy process conditioned to stay positive by using the weakly continuous bridges...

Source: http://arxiv.org/abs/1101.4184v2

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Jun 30, 2018
06/18

by
Loïc Chaumont; Gerónimo Uribe Bravo

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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...

Topics: Probability, Mathematics

Source: http://arxiv.org/abs/1405.1335

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66

Sep 21, 2013
09/13

by
Loïc Chaumont; Gerónimo Uribe Bravo

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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...

Source: http://arxiv.org/abs/0905.2155v3

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Sep 21, 2013
09/13

by
Jim Pitman; Gerónimo Uribe Bravo

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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...

Source: http://arxiv.org/abs/1011.3069v3

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7.0

Jun 29, 2018
06/18

by
Amaury Lambert; Geronimo Uribe Bravo

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We call a \emph{comb} a map $f:I\to [0,\infty)$, where $I$ is a compact interval, such that $\{f\ge \varepsilon\}$ is finite for any $\varepsilon$. A comb induces a (pseudo)-distance $\dtf$ on $\{f=0\}$ defined by $\dtf(s,t) = \max_{(s\wedge t, s\vee t)} f$. We describe the completion $\bar I$ of $\{f=0\}$ for this metric, which is a compact ultrametric space called \emph{comb metric space}. Conversely, we prove that any compact, ultrametric space $(U,d)$ without isolated points is isometric to...

Topics: Populations and Evolution, Probability, General Topology, Quantitative Biology, Mathematics

Source: http://arxiv.org/abs/1602.08246

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Sep 18, 2013
09/13

by
Clément Foucart; Gerónimo Uribe Bravo

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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.

Source: http://arxiv.org/abs/1211.3699v3

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Sep 23, 2013
09/13

by
Jean Bertoin; Geronimo Uribe Bravo

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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 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.

Source: http://arxiv.org/abs/1212.2333v1

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Sep 20, 2013
09/13

by
Maria-Emilia Caballero; Amaury Lambert; Geronimo Uribe Bravo

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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 Lamperti's 1967 representation of CSBPs in terms of spectrally positive L\'evy processes. The first proof is a direct probabilistic proof, and the second one uses approximations by discrete processes, for which the Lamperti representation is...

Source: http://arxiv.org/abs/0802.2693v2

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Sep 18, 2013
09/13

by
Josh Abramson; Jim Pitman; Nathan Ross; Gerónimo Uribe Bravo

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This article provides an overview of recent work on descriptions and properties of the convex minorant of random walks and L\'evy 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 L\'evy 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...

Source: http://arxiv.org/abs/1102.0818v1

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Jun 26, 2018
06/18

by
Ma. Emilia Caballero; José Luis Pérez Garmendia; Gerónimo Uribe Bravo

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We present a time change construction of affine processes with state-space $\mathbb{R}_+^m\times \mathbb{R}^n$. These processes were systematically studied in (Duffie, Filipovi\'c and Schachermayer, 2003) since they contain interesting classes of processes such as L\'evy processes, continuous branching processes with immigration, and of the Ornstein-Uhlenbeck type. The construction is based on a (basically) continuous functional of a multidimensional L\'evy process which implies that limit...

Topics: Mathematics, Probability

Source: http://arxiv.org/abs/1501.03122

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Sep 23, 2013
09/13

by
M. Emilia Caballero; José Luis Pérez Garmendia; Gerónimo Uribe Bravo

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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...

Source: http://arxiv.org/abs/1012.2346v4