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MAXIMUM DEGREE, CLIQUES, AND COLOURINGS

Mathew Penrose

in Random Geometric Graphs

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198506263
eISBN:
9780191707858
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506263.003.0006
Subject:
Mathematics, Probability / Statistics

This chapter is concerned with maximum degree, clique number, and chromatic number. A focusing (i.e., two-point concentration) phenomenon is demonstrated for the distribution of the maximum degree in ... More


Search for Counterparties

Darrell Duffie

in Dark Markets: Asset Pricing and Information Transmission in Over-the-Counter Markets

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691138961
eISBN:
9781400840519
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691138961.003.0003
Subject:
Economics and Finance, Financial Economics

This chapter introduces the modeling of search and random matching in large economies. The objective is to build intuition and techniques for later chapters. After some mathematical prerequisites, it ... More


Stochastic Limit Theory: An Introduction for Econometricians

James Davidson

Published in print:
1994
Published Online:
November 2003
ISBN:
9780198774037
eISBN:
9780191596117
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/0198774036.001.0001
Subject:
Economics and Finance, Econometrics

This book aims to introduce modern asymptotic theory to students and practitioners of econometrics. It falls broadly into two parts. The first half provides a handbook and reference for the ... More


Stochastic Convergence

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
1994
Published Online:
November 2003
ISBN:
9780198774037
eISBN:
9780191596117
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198774036.003.0018
Subject:
Economics and Finance, Econometrics

The modes of convergence introduced in Ch. 12 are studied in detail. Conditions for almost‐sure convergence are derived via the Borel‐Cantelli lemma. Convergence in probability is contrasted, and ... More


Convergence in Lp Norm

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
1994
Published Online:
November 2003
ISBN:
9780198774037
eISBN:
9780191596117
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198774036.003.0019
Subject:
Economics and Finance, Econometrics

This chapter looks in detail at proofs of the weak law of large numbers (convergence in probability) using the technique of establishing convergence in LP‐norm. The extension to a proof of ... More


TYPICAL VERTEX DEGREES

Mathew Penrose

in Random Geometric Graphs

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198506263
eISBN:
9780191707858
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506263.003.0004
Subject:
Mathematics, Probability / Statistics

This chapter is concerned with the empirical process of k-nearest neighbour distances for n random points, where k=k(n) is specified and is either fixed or grows with n. That is, the proportion of ... More


The Strong Law of Large Numbers

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
1994
Published Online:
November 2003
ISBN:
9780198774037
eISBN:
9780191596117
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198774036.003.0020
Subject:
Economics and Finance, Econometrics

This chapter focuses largely on methods of proof of the strong law, building on the fundamental convergence lemma. It covers Kolmogorov's three‐series theorem, strong laws for martingales, random ... More


Uniform Stochastic Convergence

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
1994
Published Online:
November 2003
ISBN:
9780198774037
eISBN:
9780191596117
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198774036.003.0021
Subject:
Economics and Finance, Econometrics

This chapter concerns random sequences of functions on metric spaces. The main issue is the distinction between convergence at all points of the space (pointwise) and uniform convergence, where limit ... More


Introduction

Stephen J. Blundell and Katherine M. Blundell

in Concepts in Thermal Physics

Published in print:
2009
Published Online:
January 2010
ISBN:
9780199562091
eISBN:
9780191718236
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562091.003.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This introductory chapter explores the concept of large numbers, showing why large numbers appear in thermal physics and explaining how to handle them. Large numbers arise in thermal physics because ... More


Statistical Causality

Jr. Henry E. Kyburg

in Science & Reason

Published in print:
1991
Published Online:
October 2011
ISBN:
9780195062533
eISBN:
9780199853038
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195062533.003.0012
Subject:
Philosophy, Philosophy of Science

An answer to the fact that it is very complex to find convincing grounds for considering in universal deterministic uniformity (much less universal causality) has been to suggest that causality is ... More


“Counting” in the French Medical Academy during the 1830s

Peter Cryle and Elizabeth Stephens

in Normality: A Critical Genealogy

Published in print:
2017
Published Online:
May 2018
ISBN:
9780226483863
eISBN:
9780226484198
Item type:
chapter
Publisher:
University of Chicago Press
DOI:
10.7208/chicago/9780226484198.003.0003
Subject:
History, History of Ideas

In 1837, a series of debates took place in the French Académie de Médecine. At issue was the relevance or otherwise of “numerical method” to medicine: why count and tabulate illnesses, treatments and ... More


Mean value—estimation and a few properties

Steve Selvin

in The Joy of Statistics: A Treasury of Elementary Statistical Tools and their Applications

Published in print:
2019
Published Online:
May 2019
ISBN:
9780198833444
eISBN:
9780191872280
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198833444.003.0003
Subject:
Mathematics, Probability / Statistics, Applied Mathematics

The mean value is perhaps the most fundamental statistic and the chapter describes its properties and features that make it an important and necessary analytic tool.


Properties of random variables

M. D. Edge

in Statistical Thinking from Scratch: A Primer for Scientists

Published in print:
2019
Published Online:
October 2019
ISBN:
9780198827627
eISBN:
9780191866463
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198827627.003.0006
Subject:
Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

In this chapter, the behavior of random variables is summarized using the concepts of expectation, variance, and covariance. The expectation is a measurement of the location of a random variable’s ... More


Introduction

Jochen Rau

in Statistical Physics and Thermodynamics: An Introduction to Key Concepts

Published in print:
2017
Published Online:
November 2017
ISBN:
9780199595068
eISBN:
9780191844300
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780199595068.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Statistical mechanics concerns the transition from the microscopic to the macroscopic realm. On a macroscopic scale new phenomena arise that have no counterpart in the microscopic world. For example, ... More


Application to the Uniform Laws of Large Numbers for Dependent Processes

Florence Merlevède, Magda Peligrad, and Sergey Utev

in Functional Gaussian Approximation for Dependent Structures

Published in print:
2019
Published Online:
April 2019
ISBN:
9780198826941
eISBN:
9780191865961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198826941.003.0016
Subject:
Mathematics, Probability / Statistics

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling ... More


Asymptotic Theory

M. Hashem Pesaran

in Time Series and Panel Data Econometrics

Published in print:
2015
Published Online:
March 2016
ISBN:
9780198736912
eISBN:
9780191800504
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198736912.003.0008
Subject:
Economics and Finance, Econometrics

Most econometric methods used in applied economics, particularly in time series econometrics, are asymptotic in the sense that they are likely to hold only when the sample size is ‘large enough’. ... More


Introduction to Probability

Therese M. Donovan and Ruth M. Mickey

in Bayesian Statistics for Beginners: a step-by-step approach

Published in print:
2019
Published Online:
July 2019
ISBN:
9780198841296
eISBN:
9780191876820
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198841296.003.0001
Subject:
Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

In this chapter, the concept of probability is introduced. The rolling of a die is an example of a random process: the face that comes up is subject to chance. In probability, the goal is to quantify ... More


Cryptography

Rutger van Santen, Djan Khoe, and Bram Vermeer

in 2030: Technology That Will Change the World

Published in print:
2010
Published Online:
November 2020
ISBN:
9780195377170
eISBN:
9780197562680
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195377170.003.0022
Subject:
Computer Science, History of Computer Science

Electronic payments, Internet shopping, and mobile communication have fundamentally changed our society and not only because digital services have made our lives more convenient. Never before in ... More


Maximal Moment Inequalities for Weakly Negatively Dependent Variables

Florence Merlevède, Magda Peligrad, and Sergey Utev

in Functional Gaussian Approximation for Dependent Structures

Published in print:
2019
Published Online:
April 2019
ISBN:
9780198826941
eISBN:
9780191865961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198826941.003.0008
Subject:
Mathematics, Probability / Statistics

The aim of this chapter is to prove Khintchine–Marcinkiewicz–Zygmund or Rosenthal-type moment inequalities for the partial sums or for the maximum of partial sums, in terms of weak dependence ... More


The Roll of the Dice

Andrew C. A. Elliott

in What Are the Chances of That?: How to Think About Uncertainty

Published in print:
2021
Published Online:
September 2021
ISBN:
9780198869023
eISBN:
9780191905490
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198869023.003.0001
Subject:
Mathematics, Applied Mathematics, Probability / Statistics

The topic of probability is introduced through analysis of games of chance, using the casino games of roulette and the dice game craps. The nature of probability is explored, including different ... More


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