scholarly journals Some exactly solvable models of urn process theory

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Philippe Flajolet ◽  
Philippe Dumas ◽  
Vincent Puyhaubert
Keyword(s):  
Large Deviation ◽  
Local Limit ◽  
Limit Laws ◽  
Coupon Collector ◽  
Exactly Solvable ◽  
Classical Models ◽  
International Audience ◽  
Contagion Model ◽  
Convergence Estimates

International audience We establish a fundamental isomorphism between discrete-time balanced urn processes and certain ordinary differential systems, which are nonlinear, autonomous, and of a simple monomial form. As a consequence, all balanced urn processes with balls of two colours are proved to be analytically solvable in finite terms. The corresponding generating functions are expressed in terms of certain Abelian integrals over curves of the Fermat type (which are also hypergeometric functions), together with their inverses. A consequence is the unification of the analyses of many classical models, including those related to the coupon collector's problem, particle transfer (the Ehrenfest model), Friedman's "adverse campaign'' and Pólya's contagion model, as well as the OK Corral model (a basic case of Lanchester's theory of conflicts). In each case, it is possible to quantify very precisely the probable composition of the urn at any discrete instant. We study here in detail "semi-sacrificial'' urns, for which the following are obtained: a Gaussian limiting distribution with speed of convergence estimates as well as a characterization of the large and extreme large deviation regimes. We also work out explicitly the case of $2$-dimensional triangular models, where local limit laws of the stable type are obtained. A few models of dimension three or greater, e.g., "autistic'' (generalized Pólya), cyclic chambers (generalized Ehrenfest), generalized coupon-collector, and triangular urns, are also shown to be exactly solvable.

scholarly journals Exactly Solvable Balanced Tenable Urns with Random Entries via the Analytic Methodology

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Basile Morcrette ◽  
Hosam M. Mahmoud
Keyword(s):  
Differential Equations ◽  
Generating Function ◽  
Probability Generating Function ◽  
Isomorphism Theorem ◽  
Systems Of Differential Equations ◽  
Exactly Solvable ◽  
Exact Distributions ◽  
International Audience ◽  
Weighted Probability

International audience This paper develops an analytic theory for the study of some Pólya urns with random rules. The idea is to extend the isomorphism theorem in Flajolet et al. (2006), which connects deterministic balanced urns to a differential system for the generating function. The methodology is based upon adaptation of operators and use of a weighted probability generating function. Systems of differential equations are developed, and when they can be solved, they lead to characterization of the exact distributions underlying the urn evolution. We give a few illustrative examples.

scholarly journals p-box: a new graph model

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Mauricio Soto ◽  
Christopher Thraves-Caro
Keyword(s):  
Graph Theory ◽  
Graph Model ◽  
Directed Path ◽  
Outerplanar Graphs ◽  
New Model ◽  
Model Based ◽  
Representative Element ◽  
International Audience ◽  
Graph Families

Graph Theory International audience In this document, we study the scope of the following graph model: each vertex is assigned to a box in ℝd and to a representative element that belongs to that box. Two vertices are connected by an edge if and only if its respective boxes contain the opposite representative element. We focus our study on the case where boxes (and therefore representative elements) associated to vertices are spread in ℝ. We give both, a combinatorial and an intersection characterization of the model. Based on these characterizations, we determine graph families that contain the model (e. g., boxicity 2 graphs) and others that the new model contains (e. g., rooted directed path). We also study the particular case where each representative element is the center of its respective box. In this particular case, we provide constructive representations for interval, block and outerplanar graphs. Finally, we show that the general and the particular model are not equivalent by constructing a graph family that separates the two cases.

scholarly journals Classes of graphs with restricted interval models

1999 ◽  
Vol Vol. 3 no. 4 ◽  
Author(s):  
Andrzej Proskurowski ◽  
Jan Arne Telle
Keyword(s):  
Maximum Clique ◽  
Interval Graphs ◽  
Induced Subgraph ◽  
Graph Theoretic ◽  
Graph Classes ◽  
International Audience ◽  
Forbidden Induced Subgraph ◽  
Nested Hierarchy ◽  
Graph Families

International audience We introduce q-proper interval graphs as interval graphs with interval models in which no interval is properly contained in more than q other intervals, and also provide a forbidden induced subgraph characterization of this class of graphs. We initiate a graph-theoretic study of subgraphs of q-proper interval graphs with maximum clique size k+1 and give an equivalent characterization of these graphs by restricted path-decomposition. By allowing the parameter q to vary from 0 to k, we obtain a nested hierarchy of graph families, from graphs of bandwidth at most k to graphs of pathwidth at most k. Allowing both parameters to vary, we have an infinite lattice of graph classes ordered by containment.

Limit theorems for first-passage times in linear and non-linear renewal theory

1984 ◽  
Vol 16 (04) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Renewal Theory ◽  
Local Limit ◽  
Linear Boundary ◽  
First Passage ◽  
Non Linear ◽  
First Time ◽  
Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

Asymptotic analysis for affine point processes with large initial intensity

2018 ◽  
Vol 17 (01) ◽  
pp. 117-143
Author(s):  
Nian Yao ◽  
Mingqing Xiao
Keyword(s):  
Asymptotic Behavior ◽  
Point Processes ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
General Structure ◽  
Large Numbers ◽  
Special Cases ◽  
Classical Models ◽  
Initial Intensity

In this paper, we consider a generalized stochastic model associated with affine point processes based on several classical models. In particular, we study the asymptotic behavior of the process when the initial intensity is large, i.e. the intensity of arriving events observed initially is considerably larger, which appears in many real applications. For our generalized model, we establish (i) the large deviation principle; (ii) the corresponding functional law of large numbers; (iii) the corresponding central limit theorem, that reflect the fundamentals of the process asymptotic behavior. Our obtained results include existing results as special cases with a more general structure.

scholarly journals Information Transmission under Random Emission Constraints

2014 ◽  
Vol 23 (6) ◽  
pp. 973-1009 ◽  
Author(s):  
FRANCIS COMETS ◽  
FRANÇOIS DELARUE ◽  
RENÉ SCHOTT
Keyword(s):  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Large Deviation Principle ◽  
Large Deviation ◽  
Total Duration ◽  
Limited Resources ◽  
Large Numbers ◽  
Coupon Collector ◽  
Selection Of ◽  
Quantities Of Interest

We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n → ∞, for the proportion of informed vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton–Watson tree with respect to the success epochs of the coupon collector problem.

scholarly journals Review on Techniques for Thermal Characterization of Graphene and Related 2D Materials

Nanomaterials ◽  
2021 ◽  
Vol 11 (11) ◽  
pp. 2787
Author(s):  
Jing Liu ◽  
Pei Li ◽  
Hongsheng Zheng
Keyword(s):  
Thermal Conductivity ◽  
Energy Transport ◽  
Large Deviation ◽  
2D Materials ◽  
Single Layer ◽  
Thermal Characterization ◽  
Great Depth ◽  
Characterization Methods ◽  
Critical Issues

The discovery of graphene and its analog, such as MoS2, has boosted research. The thermal transport in 2D materials gains much of the interest, especially when graphene has high thermal conductivity. However, the thermal properties of 2D materials obtained from experiments have large discrepancies. For example, the thermal conductivity of single layer suspended graphene obtained by experiments spans over a large range: 1100–5000 W/m·K. Apart from the different graphene quality in experiments, the thermal characterization methods play an important role in the observed large deviation of experimental data. Here we provide a critical review of the widely used thermal characterization techniques: the optothermal Raman technique and the micro-bridge method. The critical issues in the two methods are carefully revised and discussed in great depth. Furthermore, improvements in Raman-based techniques to investigate the energy transport in 2D materials are discussed.

scholarly journals Exponential bounds and tails for additive random recursive sequences

2007 ◽  
Vol Vol. 9 no. 1 (Analysis of Algorithms) ◽  
Author(s):  
Ludger Rüschendorf ◽  
Eva-Maria Schopp
Keyword(s):  
Analysis Of Algorithms ◽  
Sufficient Conditions ◽  
Random Trees ◽  
Divide And Conquer ◽  
Recursive Sequences ◽  
Recursive Structures ◽  
International Audience ◽  
Exponential Bounds ◽  
The Moment

Analysis of Algorithms International audience Exponential bounds and tail estimates are derived for additive random recursive sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise as parameters of divide and conquer type algorithms. We derive tail bounds from estimates of the Laplace transforms and of the moment sequences. For the proof we use some classical exponential bounds and some variants of the induction method. The paper generalizes results of Rösler (% \citeyearNPRoesler:91, % \citeyearNPRoesler:92) and % \citeNNeininger:05 on subgaussian tails to more general classes of additive random recursive sequences. It also gives sufficient conditions for tail bounds of the form \exp(-a t^p) which are based on a characterization of \citeNKasahara:78.

scholarly journals An optimal cardinality estimation algorithm based on order statistics and its full analysis

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Jérémie Lumbroso
Keyword(s):  
Order Statistics ◽  
Estimation Algorithm ◽  
Fine Tuning ◽  
Simple Analysis ◽  
Relative Precision ◽  
Asymptotic Regime ◽  
Cardinality Estimation ◽  
International Audience ◽  
Full Analysis

International audience Building on the ideas of Flajolet and Martin (1985), Alon et al. (1987), Bar-Yossef et al. (2002), Giroire (2005), we develop a new algorithm for cardinality estimation, based on order statistics which, according to Chassaing and Gerin (2006), is optimal among similar algorithms. This algorithm has a remarkably simple analysis that allows us to take its $\textit{fine-tuning}$ and the $\textit{characterization of its properties}$ further than has been done until now. We prove that, asymptotically, it is $\textit{strictly unbiased}$ (contrarily to Probabilistic Counting, Loglog, Hyperloglog), we verify that its relative precision is about $1/\sqrt{m-2}$ when $m$ words of storage are used, and we fully characterize the limit law of the estimates it provides, in terms of gamma distribution―-this is the first such algorithm for which the limit law has been established. We also develop a Poisson analysis for the pre-asymptotic regime. In this way, we are able to devise a complete algorithm, covering all cardinalities ranges from $0$ to very large.

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