Maximal Percentages in Pólya's Urn

Schulte-Geers, Ernst; Stadje, Wolfgang

doi:10.1239/jap/1429282614

Cited by 3 publications

(3 citation statements)

References 6 publications

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“…However, it was also independently invented in probability theory [23,33] or statistical mechanics [28,31]. Up to today it is commonly used [35,37]. For a more details on its history the interested reader is referred to [9, Chapter 1].…”

Section: The Kernel Methodsmentioning

confidence: 99%

A half-normal distribution scheme for generating functions

Wallner

2020

European Journal of Combinatorics

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We present an extension of a theorem by Michael Drmota and Michèle Soria [Images and Preimages in Random Mappings, 1997] which can be used to identify the limiting distribution for a class of combinatorial schemata. This is achieved by determining analytic and algebraic properties of the associated bivariate generating function. We give sufficient conditions implying a half-normal limiting distribution, extending the known conditions leading to either a Rayleigh, a Gaussian, or a convolution of the last two distributions. We conclude with three natural appearances of such a limiting distribution in the domain of lattice paths.

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Section: The Kernel Methodsmentioning

confidence: 99%

A half-normal distribution scheme for generating functions

Wallner

2020

European Journal of Combinatorics

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“…There are extensive theoretical developments on this scheme, see Mahmoud [4] for an excellent review. This note concerns the latest contribution due to Schulte-Geers and Stadje [6].…”

Section: Introductionmentioning

confidence: 97%

“…However, special functions (especially those for which in-built routines are widely available) defined in terms of infinite series can be referred to as closed form expressions. We also illustrate computational efficiency of the derived closed form expressions over the infinite series representations in Schulte-Geers and Stadje [6] for a wide range of values of d, r, b and t.…”

Section: Introductionmentioning

confidence: 99%

On the Distribution of Maximal Percentages in Pólya's Urn

Nadarajah¹

2016

Prob. Eng. Inf. Sci.

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Schulte-Geers and Stadje [Journal of Applied Probability, 2015, 52: 180–190] gave several closed form expressions for the exact distribution of the all-time maximal percentage in Pólya's urn model. But all these expressions corresponded to an integer parameter taking the value 1. Here, we derive much more general closed form expressions applicable for all possible values of the integer parameter. We also illustrate their computational efficiency.

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The Kernel Method for Lattice Paths Below a Line of Rational Slope

Banderier¹,

Wallner²

2019

Lattice Path Combinatorics and Applications

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We dedicate this article to the memory of Philippe Flajolet, who was and will remain a guide and a wonderful source of inspiration for so many of us. UUU AbstractWe analyse some enumerative and asymptotic properties of lattice paths below a line of rational slope. We illustrate our approach with Dyck paths under a line of slope 2/5. This answers Knuth's problem #4 from his "Flajolet lecture" during the conference "Analysis of Algorithms" (AofA'2014) in Paris in June 2014. Our approach extends the work of Banderier and Flajolet for asymptotics and enumeration of directed lattice paths to the case of generating functions involving several dominant singularities, and has applications to a full class of problems involving some "periodicities".A key ingredient in the proof is the generalization of an old trick by Knuth himself (for enumerating permutations sortable by a stack), promoted by Flajolet and others as the "kernel method". All the corresponding generating functions are algebraic, and they offer some new combinatorial identities, which can also be tackled in the A=B spirit of Wilf-Zeilberger-Petkovšek.We show how to obtain similar results for any rational slope. An interesting case is e.g. Dyck paths below the slope 2/3 (this corresponds to the so-called Duchon's club model), for which we solve a conjecture related to the asymptotics of the area below such lattice paths. Our work also gives access to lattice paths below an irrational slope (e.g. Dyck paths below y = x/ √ 2), a problem that we study in a companion article.Lattice paths below a line of rational slope 1 The "kernel method" that we mention here for functional equations in combinatorics has nothing to do with what is known as the "kernel method" or "kernel trick" in statistics or machine learning. Also, there is no integral directly related to our kernel. For sure, in our case the word kernel was chosen as its zeros will play a key role, and also, in one sense, as this kernel has in its core the full description of the problem, and its resolution.

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Maximal Percentages in Pólya's Urn

Abstract: Abstract. We show that the supremum of the successive percentages of red balls in Pólya's urn model is almost surely rational, give the set of values that are taken with positive probability and derive several exact distributional results for the all-time maximal percentage.

Cited by 3 publications

References 6 publications

A half-normal distribution scheme for generating functions

A half-normal distribution scheme for generating functions

On the Distribution of Maximal Percentages in Pólya's Urn

The Kernel Method for Lattice Paths Below a Line of Rational Slope

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