scholarly journals Additive tree functionals with small toll functions and subtrees of random trees

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Stephan Wagner
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Trees ◽  
Divide And Conquer ◽  
Main Motivation ◽  
Additive Tree ◽  
International Audience ◽  
Number Of Leaves ◽  
Labelled Trees

International audience Many parameters of trees are additive in the sense that they can be computed recursively from the sum of the branches plus a certain toll function. For instance, such parameters occur very frequently in the analysis of divide-and-conquer algorithms. Here we are interested in the situation that the toll function is small (the average over all trees of a given size $n$ decreases exponentially with $n$). We prove a general central limit theorem for random labelled trees and apply it to a number of examples. The main motivation is the study of the number of subtrees in a random labelled tree, but it also applies to classical instances such as the number of leaves.

Gibbs' Measures on Combinatorial Objects and the Central Limit Theorem for an Exponential Family of Random Trees

1987 ◽  
Vol 1 (1) ◽  
pp. 47-59 ◽  
Author(s):  
Michael Steele
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Exponential Family ◽  
Gibbs Measures ◽  
Random Trees ◽  
Exponential Families ◽  
Combinatorial Objects ◽  
Uniform Model ◽  
Number Of Leaves

A model for random trees is given which provides an embedding of the uniform model into an exponential family whose natural parameter is the expected number of leaves. The model is proved to be analytically and computationally tractable. In particular, a central limit theorem (CLT) for the number of leaves of a random tree is given which extends and sharpens Rényi's CLT for the uniform model. The method used is general and is shown to provide tractable exponential families for a variety of combinatorial objects.

scholarly journals Constrained exchangeable partitions

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Alexander Gnedin
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Partitions ◽  
Type Representation ◽  
International Audience ◽  
De Finetti ◽  
Infinite Set ◽  
Special Case

International audience For a class of random partitions of an infinite set a de Finetti-type representation is derived, and in one special case a central limit theorem for the number of blocks is shown.

scholarly journals Point process stabilization methods and dimension estimation

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Point Process ◽  
Central Limit ◽  
Point Processes ◽  
Dimension Estimation ◽  
Stabilization Methods ◽  
International Audience

International audience We provide an overview of stabilization methods for point processes and apply these methods to deduce a central limit theorem for statistical estimators of dimension.

scholarly journals Random Inscribing Polytopes

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Ross M. Richardson ◽  
Van H. Vu ◽  
Lei Wu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Poisson Process ◽  
Central Limit ◽  
Convex Bodies ◽  
Higher Moments ◽  
Random Polytopes ◽  
International Audience

International audience For convex bodies $K$ with $\mathcal{C}^2$ boundary in $\mathbb{R}^d$, we provide results on the volume of random polytopes with vertices chosen along the boundary of $K$ which we call $\textit{random inscribing polytopes}$. In particular, we prove results concerning the variance and higher moments of the volume, as well as show that the random inscribing polytopes generated by the Poisson process satisfy central limit theorem.

scholarly journals The degree distribution in unlabelled $2$-connected graph families

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Veronika Kraus
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Connected Graph ◽  
Degree Distribution ◽  
Random Variable ◽  
Expected Value ◽  
Asymptotically Linear ◽  
International Audience ◽  
Graph Families

International audience We study the random variable $X_n^k$, counting the number of vertices of degree $k$ in a randomly chosen $2$-connected graph of given families. We prove a central limit theorem for $X_n^k$ with expected value $\mathbb{E}X_n^k \sim \mu_kn$ and variance $\mathbb{V}X_n^k \sim \sigma_k^2n$, both asymptotically linear in $n$, for both rooted and unrooted unlabelled $2$-connected outerplanar or series-parallel graphs.

scholarly journals Fluctuations of central measures on partitions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Pierre-Loïc Mèliot
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Symmetric Group ◽  
Central Limit ◽  
Symmetric Groups ◽  
Polynomial Functions ◽  
Young Diagrams ◽  
Random Partitions ◽  
Infinite Symmetric Group ◽  
International Audience

International audience We study the fluctuations of models of random partitions $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ stemming from the representation theory of the infinite symmetric group. Using the theory of polynomial functions on Young diagrams, we establish a central limit theorem for the values of the irreducible characters $χ ^λ$ of the symmetric groups, with $λ$ taken randomly according to the laws $\mathbb{P}_n,ω$ . This implies a central limit theorem for the rows and columns of the random partitions, and these ``geometric'' fluctuations of our models can be recovered by relating central measures on partitions, generalized riffle shuffles, and Brownian motions conditioned to stay in a Weyl chamber. Nous étudions les fluctuations de modèles de partitions aléatoires $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ issus de la théorie des représentations du groupe symétrique infini. En utilisant la théorie des fonctions polynomiales sur les diagrammes de Young, nous établissons un théorème central limite pour les valeurs des caractères irréductibles $χ ^λ$ des groupes symétriques, avec $λ$ pris aléatoirement suivant les lois $\mathbb{P}_n,ω$ . Ceci implique un théorème central limite pour les lignes et les colonnes des partitions aléatoires, et ces fluctuations ``géométriques'' de nos modèles peuvent être retrouvées en reliant les mesures centrales sur les partitions, les battages généralisés de cartes, et les mouvements browniens conditionnés à rester dans une chambre de Weyl.

scholarly journals Kerov's central limit theorem for Schur-Weyl and Gelfand measures (extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Pierre-Loïc Méliot
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Gaussian Process ◽  
Central Limit ◽  
Integer Partitions ◽  
Nous Montrons ◽  
International Audience ◽  
The One

International audience We show that the shapes of integer partitions chosen randomly according to Schur-Weyl measures of parameter $\alpha =1/2$ and Gelfand measures satisfy Kerov's central limit theorem. Thus, there is a gaussian process $\Delta$ such that under Plancherel, Schur-Weyl or Gelfand measures, the deviations $\Delta_n(s)=\lambda _n(\sqrt{n} s)-\sqrt{n} \lambda _{\infty}^{\ast}(s)$ converge in law towards $\Delta (s)$, up to a translation along the $x$-axis in the case of Schur-Weyl measures, and up to a factor $\sqrt{2}$ and a deterministic remainder in the case of Gelfand measures. The proofs of these results follow the one given by Ivanov and Olshanski for Plancherel measures; hence, one uses a "method of noncommutative moments''. Nous montrons que les formes des partitions d'entiers choisies aléatoirement sous les mesures de Schur-Weyl de paramètre $\alpha =1/2$ et sous les mesures de Gelfand obéissent au théorème central limite de Kerov. Ainsi, il existe un processus gaussien $\Delta$ tel que sous les mesures de Plancherel, de Schur-Weyl ou de Gelfand, les déviations $\Delta_n(s)=\lambda _n(\sqrt{n} s)-\sqrt{n} \lambda _{\infty}^{\ast}(s)$ convergent en loi vers $\Delta (s)$, à une translation près le long de l'axe des abscisses pour les mesures de Schur-Weyl, et à un facteur $\sqrt{2}$ et un reste déterministe près dans le cas des mesures de Gelfand. Les preuves de ces résultats suivent celle donnée par Ivanov et Olshanski pour les mesures de Plancherel; ainsi, on utilise une "méthode de moments non commutatifs''.

scholarly journals A Central Limit Theorem for Almost Local Additive Tree Functionals

Algorithmica ◽  
2019 ◽  
Vol 82 (3) ◽  
pp. 642-679
Author(s):  
Dimbinaina Ralaivaosaona ◽  
Matas Šileikis ◽  
Stephan Wagner
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Additive Tree

scholarly journals A localized Erdős-Kac theorem

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Anup B Dixit ◽  
M Ram Murty
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Independent Random Variables ◽  
Prime Divisors ◽  
International Audience ◽  
Increasing Function

International audience Let ω_y (n) be the number of distinct prime divisors of n not exceeding y. If y_n is an increasing function of n such that log y_n = o(log n), we study the distribution of ω_{y_n} (n) and establish an analog of the Erdős-Kac theorem for this function. En route, we also prove a variant central limit theorem for random variables, which are not necessarily independent, but are well approximated by independent random variables.

Sign in / Sign up

Export Citation Format

Share Document