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CONNECTIVITY AND THE NUMBER OF COMPONENTS

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/9780198506263.003.0013
Subject:
Mathematics, Probability / Statistics

This chapter is mainly concerned with the connectivity threshold T on n random points (i.e., the smallest r at which G(n,r) is connected). The results for this threshold are similar to those for the ... 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


Probability and Statistical Distributions

Željko Ivezi, Andrew J. Connolly, Jacob T. VanderPlas, Alexander Gray, Željko Ivezi, Andrew J. Connolly, Jacob T. VanderPlas, and Alexander Gray

in Statistics, Data Mining, and Machine Learning in Astronomy: A Practical Python Guide for the Analysis of Survey Data

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691151687
eISBN:
9781400848911
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691151687.003.0003
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter reviews notation and basic concepts in probability and statistics. The coverage of various topics cannot be complete, and it is aimed at concepts needed to understand material covered in ... More


PRO-SUBJECTIVE APPROACH LOSES AS SAMPLING THEORY GAINS GROUND

Shoutir Kishore Chatterjee

in Statistical Thought: A Perspective and History

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198525318
eISBN:
9780191711657
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198525318.003.0008
Subject:
Mathematics, Probability / Statistics

In the first half of the 19th century, Laplace, Gauss, and a number of other contributors enriched statistical thought. The quest for a suitable model for the distribution of observational errors led ... More


Weak Convergence in a Function Space

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:
10.1093/0198774036.003.0027
Subject:
Economics and Finance, Econometrics

This chapter applies the theory of Ch. 26 to the case of the space C of continuous functions on the unit interval. It is shown how to assign probability measures to C, and Weiner measure (Brownian ... More


The early history of cumulants and the Gram–Charlier series 79

A. Hald

in Thiele: Pioneer in Statistics

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198509721
eISBN:
9780191709197
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509721.003.0007
Subject:
Mathematics, Probability / Statistics

This chapter presents a reprint of part of Hald (2000a), containing a detailed discussion of Thiele's halfinvariants and their use in series expansions. Section 2 describes how Poisson, Bessel, and ... More


CLTs for Dependent Processes

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:
10.1093/0198774036.003.0024
Subject:
Economics and Finance, Econometrics

This chapter deals with the central limit theorem (CLT) for dependent processes. As with the law of large numbers, the focus is on near‐epoch dependent and mixing processes, and array versions of the ... More


Universality and the continuum limit

Jean Zinn-Justin

in Phase Transitions and Renormalization Group

Published in print:
2007
Published Online:
January 2010
ISBN:
9780199227198
eISBN:
9780191711107
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199227198.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter discusses the related questions of universality and macroscopic continuum limit in random systems with a large number of degrees of freedom. It first explains the notion of universality ... More


Gaussian Approximation via Martingale Methods

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/9780198826941.003.0004
Subject:
Mathematics, Probability / Statistics

Gordin’s seminal paper (1969) initiated a line of research in which limit theorems for stationary sequences are proved via appropriate approximations by stationary martingale difference sequences ... More


Ergodic Theory

Robert Alicki and Mark Fannes

in Quantum Dynamical Systems

Published in print:
2001
Published Online:
February 2010
ISBN:
9780198504009
eISBN:
9780191708503
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504009.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and ... More


Moment Inequalities and Gaussian Approximation for Mixing Sequences

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/9780198826941.003.0006
Subject:
Mathematics, Probability / Statistics

Although the “mixing assumption” is nowadays considered as a rather restrictive condition, it is a powerful tool, which possesses strong properties such as preservation under functional transform, ... More


The central limit theorem for geodesics

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion: An Introduction

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199654109
eISBN:
9780191745676
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199654109.003.0008
Subject:
Mathematics, Mathematical Physics

This chapter provides a proof of the Sinai central limit theorem, generalized to the case of a cofinite and geometrically finite Kleinian group. This theorem shows that, asymptotically, geodesics ... 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


The Patterns of Large Numbers

Steven J. Osterlind

in The Error of Truth: How History and Mathematics Came Together to Form Our Character and Shape Our Worldview

Published in print:
2019
Published Online:
January 2019
ISBN:
9780198831600
eISBN:
9780191869532
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198831600.003.0004
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter advances the historical context for quantification by describing the climate of the day—social, cultural, political, and intellectual—as fraught with disquieting influences. Forces ... More


Reversible Markov Chains

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/9780198826941.003.0014
Subject:
Mathematics, Probability / Statistics

This chapter is dedicated to the Gaussian approximation of a reversible Markov chain. Regarding this problem, the coefficients of dependence for reversible Markov chains are actually the covariances ... More


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


Moment Inequalities and Gaussian Approximation for Martingales

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/9780198826941.003.0002
Subject:
Mathematics, Probability / Statistics

The aim of this chapter is to present useful tools for analyzing the asymptotic behavior of partial sums associated with dependent sequences, by approximating them with martingales. We start by ... More


Hyperbolic Dynamics and Brownian Motion: An Introduction

Jacques Franchi and Yves Le Jan

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199654109
eISBN:
9780191745676
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199654109.001.0001
Subject:
Mathematics, Mathematical Physics

The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) ... More


Probability and Random Processes

Stefan Thurner, Rudolf Hanel, and Peter Klimekl

in Introduction to the Theory of Complex Systems

Published in print:
2018
Published Online:
November 2018
ISBN:
9780198821939
eISBN:
9780191861062
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821939.003.0002
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Phenomena, systems, and processes are rarely purely deterministic, but contain stochastic,probabilistic, or random components. For that reason, a probabilistic descriptionof most phenomena is ... More


Gaussian Approximation under Asymptotic Negative Dependence

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/9780198826941.003.0009
Subject:
Mathematics, Probability / Statistics

Here we introduce the notion of asymptotic weakly associated dependence conditions, the practical applications of which will be discussed in the next chapter. The theoretical importance of this class ... More


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