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23
Functional Limit Theorems For Multitype Branching Processes And Generalized Pólya Urns
 APPL
, 2004
"... A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. Using the ..."
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Cited by 73 (13 self)
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A functional limit theorem is proved for multitype continuous time Markov branching processes. As consequences, we obtain limit theorems for the branching process stopped by some stopping rule, for example when the total number of particles reaches a given level. Using the
Limit theorems for triangular urn schemes
 PROB. THEORY RELATED FIELDS
, 2005
"... We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depen ..."
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Cited by 27 (2 self)
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We study a generalized Pólya urn with balls of two colours and a triangular replacement matrix; the urn is not required to be balanced. We prove limit theorems describing the asymptotic distribution of the composition of the urn after a long time. Several different types of asymptotics appear, depending on the ratio of the diagonal elements in the replacement matrix; the limit laws include normal, stable and MittagLeffler distributions as well as some less familiar ones. The results are in some cases similar to, but in other cases strikingly different from, the results for irreducible replacement matrices.
Limit laws for local counters in random binary search trees
 Random Structures and Algorithms
, 1991
"... Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for rndependent random variables. Examples include: the number of leaves (L a), the number of nodes with k de ..."
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Cited by 21 (2 self)
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Limit laws for several quantities in random binary search trees that are related to the local shape of a tree around each node can be obtained very simply by applying central limit theorems for rndependent random variables. Examples include: the number of leaves (L a), the number of nodes with k descendants (k fixed), the number of nodes with no left child, the number of nodes with k left descendants. Some of these results can also be obtained via the theory of urn models, but the present method seems easier to apply.
Asymptotic Degree Distribution In Random Recursive Trees
 Random Structures & Algorithms
, 2005
"... The distributions of vertex degrees in random recursive trees and random plane recursive trees are shown to be asymptotically normal. Formulas are given for the asymptotic variances and covariances of the number of vertices with given outdegrees. We also give functional limit theorems for the ev ..."
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Cited by 18 (3 self)
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The distributions of vertex degrees in random recursive trees and random plane recursive trees are shown to be asymptotically normal. Formulas are given for the asymptotic variances and covariances of the number of vertices with given outdegrees. We also give functional limit theorems for the evolution as vertices are added.
An algebraic approach of Pólya processes
 Submitted
"... Abstract. Pólya processes are natural generalizations of Pólya–Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provi ..."
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Cited by 11 (3 self)
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Abstract. Pólya processes are natural generalizations of Pólya–Eggenberger urn models. This article presents a new approach of their asymptotic behaviour via moments, based on the spectral decomposition of a suitable finite difference transition operator on polynomial functions. Especially, it provides new results for large processes (a Pólya process is called small when1isasimple eigenvalue of its replacement matrix and when any other eigenvalue has a real part ≤ 1/2; otherwise, it is called large). Résumé. Les processus de Pólya sont une généralisation naturelle des modèles d’urnes de Pólya–Eggenberger. Cet article présente une nouvelle approche de leur comportement asymptotique via les moments, basée sur la décomposition spectrale d’un opérateur aux différences finies sur des espaces de polynômes. En particulier, elle fournit de nouveaux résultats sur les grands processus (un processus de Pólya est dit petit lorsque 1 est valeur propre simple de sa matrice de remplacement et lorsque toutes les autres valeurs propres ont une partie réelle ≤ 1/2; sinon, on dit qu’il est grand).
Fringe Analysis Revisited
"... Fringe analysis is a technique used to study the average behavior of search trees. In this paper we survey the main results regarding this technique, and we improve a previous asymptotic theorem. At the same time we present new developments and applications of the theory which allow improvements in ..."
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Cited by 11 (5 self)
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Fringe analysis is a technique used to study the average behavior of search trees. In this paper we survey the main results regarding this technique, and we improve a previous asymptotic theorem. At the same time we present new developments and applications of the theory which allow improvements in several bounds on the behavior of search trees. Our examples cover binary search trees, AVL trees, 23 trees, and Btrees. Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity ]: Nonnumerical Algorithms and Problems  computations on discrete structures; sorting and searching; E.1 [Data Structures]; trees. Contents 1 Introduction 2 2 The Theory of Fringe Analysis 4 3 Weakly Closed Collections 9 4 Including the Level Information 11 5 Fringe Analysis, Markov Chains, and Urn Processes 13 This work was partially funded by Research Grant FONDECYT 930765. email: rbaeza@dcc.uchile.cl 1 Introduction Search trees are one of the most used data structures t...
Classification of large PólyaEggenberger urns with regard to their asymptotics
 DISCRETE MATHEMATICS AND THEORETICAL COMPUTER SCIENCE (SUBM.)
"... This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having “large ” eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of t ..."
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Cited by 10 (4 self)
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This article deals with Pólya generalized urn models with constant balance in any dimension. It is based on the algebraic approach of Pouyanne (2005) and classifies urns having “large ” eigenvalues in five classes, depending on their almost sure asymptotics. These classes are described in terms of the spectrum of the urn’s replacement matrix and examples of each case are treated. We study the cases of socalled cyclic urns in any dimension and mary search trees for m ≥ 27.
Congruence properties of depths in some random trees
, 2005
"... Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n → ∞. The same is true for the number of vertices of depth divisible by m for m = 3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m ≥ 7 ..."
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Cited by 4 (0 self)
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Consider a random recusive tree with n vertices. We show that the number of vertices with even depth is asymptotically normal as n → ∞. The same is true for the number of vertices of depth divisible by m for m = 3, 4 or 5; in all four cases the variance grows linearly. On the other hand, for m ≥ 7, the number is not asymptotically normal, and the variance grows faster than linear in n. The case m = 6 is intermediate: the number is asymptotically normal but the variance is of order nlog n. This is a simple and striking example of a type of phase transition that has been observed by other authors in several cases. We prove, and perhaps explain, this nonintuitive behavious using a translation to a generalized Pólya urn. Similar results hold for a random binary search tree; now the number of vertices of depth divisible by m is asymptotically normal for m ≤ 8 but not for m ≥ 9, and the variance grows linearly in the first case both faster in the second. (There is no intermediate case.) In contrast, we show that for conditioned Galton–Watson trees, including random labelled trees and random binary trees, there is no such phase transition: the number is asymptotically normal for every m.
Fully Analyzing an Algebraic Pólya Urn Model
"... Abstract. This paper introduces and analyzes a particular class of Pólya urns: balls are of two colors, can only be added (the urns are said to be additive) and at every step the same constant number of balls is added, thus only the color compositions varies (the urns are said to be balanced). These ..."
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Cited by 2 (0 self)
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Abstract. This paper introduces and analyzes a particular class of Pólya urns: balls are of two colors, can only be added (the urns are said to be additive) and at every step the same constant number of balls is added, thus only the color compositions varies (the urns are said to be balanced). These properties make this class of urns ideally suited for analysis from an “analytic combinatorics ” pointofview, following in the footsteps of Flajolet et al. [4]. Through an algebraic generating function to which we apply a multiple coalescing saddlepoint method, we are able to give precise asymptotic results for the probability distribution of the composition of the urn, as well as local limit law and large deviation bounds.