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Random subgraph counts and U-statistics: multivariate normal approximation via exchangeable pairs and embedding

Part of: Limit theorems Distribution theory Graph theory

Published online by Cambridge University Press:  14 July 2016

Gesine Reinert  and
Adrian Röllin
Gesine Reinert*
Affiliation:
University of Oxford
Adrian Röllin*
Affiliation:
National University of Singapore
*
∗Postal address: Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK.
∗∗Postal address: Department of Statistics and Applied Probability, National University of Singapore, 6 Science Drive 2, Singapore 117546. Email address: [email protected]
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Abstract

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In Reinert and Röllin (2009) a new approach - called the ‘embedding method’ - was introduced, which allows us to make use of exchangeable pairs for normal and multivariate normal approximations with Stein's method in cases where the corresponding couplings do not satisfy a certain linearity condition. The key idea is to embed the problem into a higher-dimensional space in such a way that the linearity condition is then satisfied. Here we apply the embedding to U-statistics as well as to subgraph counts in random graphs.

Keywords

Multivariate normal distributionU-statisticsrandom graph statisticsStein's methodexchangeable pair

MSC classification

Secondary: 60F05: Central limit and other weak theorems 62E17: Approximations to distributions (nonasymptotic) 05C80: Random graphs
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 378 - 393
Copyright
Copyright © Applied Probability Trust 2010 

Footnotes

Supported in part by BBsrc and EPSRC Through OCISB.

Supported in part by the Swiss National Science Foundation project PBZH2—117033.

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