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19
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 67 (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 25 (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 20 (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.
Analytic Urns
 March
, 2003
"... This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matri ..."
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Cited by 20 (1 self)
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This article describes a purely analytic approach to urn models of the generalized or extended PólyaEggenberger type, in the case of two types of balls and constant "balance", i.e., constant row sum. (Under such models, an urn may contain balls of either of two colours and a fixed 2 × 2matrix determines the replacement policy when a ball is drawn and its colour is observed.) The treatment starts from a quasilinear firstorder partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences are new representations for the probability distribution of the urn's composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limits, and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x = 1. Several urn models, including a classical one associated with balanced trees (23 trees and fringebalanced search trees) and related to a previous study of Panholzer and Prodinger as well as all urns of balance 1 or 2, are shown to admit of explicit representations in terms of Weierstraß elliptic functions. Other consequences include a unification of earlier studies of these models and the detection of stable laws in certain classes of urns with an offdiagonal entry equal to zero.
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 15 (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.
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 12 (6 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...
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 12 (5 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).
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 (5 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.
Applications of Steins method in the analysis of random binary search trees. Steins method and Applications
 Institute for Mathematical Sciences Lecture Notes Series
, 2005
"... Abstract. Under certain conditions, sums of functions of subtrees of a random binary search tree are asymptotically normal. We show how Stein’s method can be applied to study these random trees, and survey methods for obtaining limit laws for such functions of subtrees. ..."
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Cited by 1 (0 self)
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Abstract. Under certain conditions, sums of functions of subtrees of a random binary search tree are asymptotically normal. We show how Stein’s method can be applied to study these random trees, and survey methods for obtaining limit laws for such functions of subtrees.
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 1 (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.