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Random Geometric Graphs$
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Mathew Penrose

Print publication date: 2003

Print ISBN-13: 9780198506263

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506263.001.0001

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PROBABILISTIC INGREDIENTS

PROBABILISTIC INGREDIENTS

Chapter:
(p.22) 2 PROBABILISTIC INGREDIENTS
Source:
Random Geometric Graphs
Author(s):

Mathew Penrose (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506263.003.0002

This chapter derives some probabilistic results for later use. A dependency graph for a family of random variables is a deterministic graph with vertex-set identified with the random variables and edges corresponding to dependencies between variables. Univariate and multivariate Poisson approximation theorems, and normal approximation theorems, are derived for sums of variables having a dependency graph using Stein's method. Random variables given as sums of martingale differences are also considered. For such variables, both concentration inequalities and a central limit theorem are given. Finally, a ‘de-Poissonization’ result is given which is to be used for extending central limit theorems for G(N(n),r) (with N(n) Poisson with mean n) to analogous results for G(n,r).

Keywords:   dependency graph, Poisson approximation, normal approximation, martingale differences, Stein's method, concentration inequalities, de-Poissonization

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