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Valeurs principales associées aux temps locaux browniens
TL;DR: In this article, the valeur principale de Cauchy des temps locaux du mouvement brownien lineaire is examined, and a loi for diverses valeurs du temps is presented.
Abstract: On etudie la valeur principale de Cauchy des temps locaux du mouvement brownien lineaire. En particulier, on donne sa loi pour diverses valeurs du temps; les calculs utilisent la theorie des excursions du mouvement brownien et menent a des resultats simples, relies a une formule de P. Levy
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01 Jan 2006
TL;DR: In this paper, the Brownian forest and the additive coalescent were constructed for random walks and random forests, respectively, and the Bessel process was used for random mappings.
Abstract: Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.
1,371 citations
01 Oct 1991
TL;DR: In this paper, the authors discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes, aimed at theoretical probabilists.
Abstract: INTRODUCTION Many different models of random trees have arisen in a variety of applied setting, and there is a large but scattered literature on exact and asymptotic results for particular models. For several years I have been interested in what kinds of “general theory” (as opposed to ad hoc analysis of particular models) might be useful in studying asymptotics of random trees. In this paper, aimed at theoretical probabilists, I discuss aspects of this incipient general theory which are most closely related to topics of current interest in theoretical stochastic processes. No prior knowledge of this subject is assumed: the paper is intended as an introduction and survey. To give the really big picture in a paragraph, consider a tree on n vertices. View the vertices as points in abstract (rather than d -dimensional) space, but let the edges have length (= 1, as a default) so that there is metric structure: the distance between two vertices is the length of the path between them. Consider the average distance between pairs of vertices. As n → ∞ this average distance could stay bounded or could grow as order n , but almost all natural random trees fall into one of two categories. In the first (and larger) category, the average distance grows as order logn. This category includes supercritical branching processes, and most “Markovian growth” models such as those occurring in the analysis of algorithms. This paper is concerned with the second category, in which the average distance grows as order n ½ .
507 citations
TL;DR: In this paper, a review of known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables is presented.
Abstract: This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.
304 citations
TL;DR: The number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels when edges are assignedrandom labels.
Abstract: We study random cutting down of a rooted tree and show that the number of cuts is equal (in distribution) to the number of records in the tree when edges (or vertices) are assigned random labels.
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128 citations
Cites background from "Valeurs principales associées aux t..."
...) In particular, we see that we have two different descriptions of the limit of H(Tn), and thus [6]...
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TL;DR: In this paper, a Bernoulli excursion is used to find the distribution, the moments, and the asymptotic distribution of the random variable ow, defined by 2nwo = + 1r? + S" + 72n.
Abstract: This paper is concerned with a random walk process {l , n1i, . , ri} in which 772n = 70=0 and -i 0 for i = 1, 2, -., 2n. This process is called a Bernoulli excursion. The main object is to find the distribution, the moments, and the asymptotic distribution of the random variable ow, defined by 2nwo = + 1r? + S" + 72n. The results derived have various applications in the theory of probability, including random graphs, tournaments and order statistics. BROWNIAN EXCURSION; RANDOM GRAPHS; ORDER STATISTICS; TOURNAMENTS
116 citations
Cites result from "Valeurs principales associées aux t..."
...For other results concerning w+we refer to Cohen and Hooghiemstra [4], Biane and Yor [2], and Groeneboom [10]....
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