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Independent process approximations for random combinatorial structures
 ADVANCES IN MATHEMATICS
, 1994
"... Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to compare the combinatorial structure directly to the independ ..."
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Cited by 38 (8 self)
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Many random combinatorial objects have a component structure whose joint distribution is equal to that of a process of mutually independent random variables, conditioned on the value of a weighted sum of the variables. It is interesting to compare the combinatorial structure directly to the independent discrete process, without renormalizing. The quality of approximation can often be conveniently quantified in terms of total variation distance, for functionals which observe part, but not all, of the combinatorial and independent processes. Among the examples are combinatorial assemblies (e.g., permutations, random mapping functions, and partitions of a set), multisets (e.g, polynomials over a finite field, mapping patterns and partitions of an integer), and selections (e.g., partitions of an integer into distinct parts, and squarefree polynomials over finite fields). We consider issues common to all the above examples, including equalities and upper bounds for total variation distances, existence of limiting processes, heuristics for good approximations, the relation to standard generating functions, moment formulas and recursions for computing densities, refinement to the process which counts the number of parts of each possible type, the effect of further conditioning on events of moderate probability, large deviation theory and nonuniform measures on combinatorial objects, and the possibility of getting useful results by overpowering the conditioning.
Limit theorems for combinatorial structures via discrete process approximations
 RANDOM STRUCTURES AND ALGORITHMS
, 1992
"... Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide e ..."
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Cited by 21 (2 self)
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Discrete functional limit theorems, which give independent process approximations for the joint distribution of the component structure of combinatorial objects such as permutations and mappings, have recently become available. In this article, we demonstrate the power of these theorems to provide elementary proofs of a variety of new and old limit theorems, including results previously proved by complicated analytical methods. Among the examples we treat are Brownian motion limit theorems for the cycle counts of a random permutation or the component counts of a random mapping, a Poisson limit law for the core of a random mapping, a generalization of the ErdosTurin Law for the logorder of a random permutation and the smallest component size of a random permutation, approximations to the joint laws of the smallest cycle sizes of a random mapping, and a limit distribution for the difference between the total number of cycles and the number of
Distance estimates for dependent superpositions of point processes
 Stochastic Processes and their Applications 115
"... Abstract In this article, superpositions of possibly dependent point processes on a general space X are considered. Using Stein's method for Poisson process approximation, an estimate is given for the Wasserstein distance d 2 between the distribution of such a superposition and an appropriate ..."
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Cited by 6 (0 self)
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Abstract In this article, superpositions of possibly dependent point processes on a general space X are considered. Using Stein's method for Poisson process approximation, an estimate is given for the Wasserstein distance d 2 between the distribution of such a superposition and an appropriate Poisson process distribution. This estimate is compared to a modern version of Grigelionis' theorem, and to results of
A new metric between distributions of point processes
, 2007
"... Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric ¯ d1 that combines positional differences of points under a closest match with the relati ..."
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Cited by 2 (1 self)
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Most metrics between finite point measures currently used in the literature have the flaw that they do not treat differing total masses in an adequate manner for applications. This paper introduces a new metric ¯ d1 that combines positional differences of points under a closest match with the relative difference in total mass in a way that fixes this flaw. A comprehensive collection of theoretical results about ¯ d1 and its induced Wasserstein metric ¯d2 for point process distributions are given, including examples of useful ¯ d1Lipschitz continuous functions, ¯ d2 upper bounds for Poisson process approximation, and ¯ d2 upper and lower bounds between distributions of point processes of i.i.d. points. Furthermore, we present a statistical test for multiple point pattern data that demonstrates the potential of ¯ d1 in applications.
Poisson Process Approximation: From Palm Theory to Stein's Method
"... Abstract: This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigrationdeath process and Palm theory. The latter approach also enables us to define local dependence of point proc ..."
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Abstract: This exposition explains the basic ideas of Stein's method for Poisson random variable approximation and Poisson process approximation from the point of view of the immigrationdeath process and Palm theory. The latter approach also enables us to define local dependence of point processes
Normal Approximation by Stein's Method
"... The aim of this paper is to give an overview of Stein's method, which has turned out to be a powerful tool for estimating the error in normal, Poisson and other approximations, especially for sums of dependent random variables. We focus on the normal approximation of random variables posessing ..."
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The aim of this paper is to give an overview of Stein's method, which has turned out to be a powerful tool for estimating the error in normal, Poisson and other approximations, especially for sums of dependent random variables. We focus on the normal approximation of random variables posessing decompositions of Barbour, Karonski, and Rucinski (1989), which are particularly useful in combinatorial structures, where there is no natural ordering of the summands. We highlight two applications: Nash equilibria and linear rank statistics. 1
© Institute of Mathematical Statistics, 2004 STEIN’S METHOD, PALM THEORY AND POISSON
"... The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is ..."
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The framework of Stein’s method for Poisson process approximation is presented from the point of view of Palm theory, which is used to construct Stein identities and define local dependence. A general result (Theorem 2.3) in Poisson process approximation is proved by taking the local approach. It is obtained without reference to any particular metric, thereby allowing wider applicability. A Wasserstein pseudometric is introduced for measuring the accuracy of point process approximation. The pseudometric provides a generalization of many metrics used so far, including the total variation distance for random variables and the Wasserstein metric for processes as in Barbour and Brown [Stochastic Process. Appl. 43 (1992) 9–31]. Also, through the pseudometric, approximation for certain point processes on a given carrier space is carried out by lifting it to one on a larger space, extending an idea of Arratia, Goldstein and Gordon [Statist. Sci. 5 (1990) 403–434]. The error bound in the general result is similar in form to that for Poisson approximation. As it yields the Stein factor 1/λ as in Poisson approximation, it provides good approximation, particularly in cases where λ is large. The general result is applied to a number of problems including Poisson process modeling of rare words in a DNA sequence. 1. Introduction. Poisson
coupling
, 2005
"... Compound Poisson process approximation for locally dependent real valued random variables via a new ..."
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Compound Poisson process approximation for locally dependent real valued random variables via a new