## 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 L_{p} 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