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2019 Multivariate approximation in total variation using local dependence
A.D. Barbour, A. Xia
Electron. J. Probab. 24: 1-35 (2019). DOI: 10.1214/19-EJP284


We establish two theorems for assessing the accuracy in total variation of multivariate discrete normal approximation to the distribution of an integer valued random vector $W$. The first is for sums of random vectors whose dependence structure is local. The second applies to random vectors $W$ resulting from integrating the ${\mathbb Z}^d$-valued marks of a marked point process with respect to its ground process. The error bounds are of magnitude comparable to those given in [Rinott & Rotar (1996)], but now with respect to the stronger total variation distance. Instead of requiring the summands to be bounded, we make third moment assumptions. We demonstrate the use of the theorems in four applications: monochrome edges in vertex coloured graphs, induced triangles and $2$-stars in random geometric graphs, the times spent in different states by an irreducible and aperiodic finite Markov chain, and the maximal points in different regions of a homogeneous Poisson point process.


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A.D. Barbour. A. Xia. "Multivariate approximation in total variation using local dependence." Electron. J. Probab. 24 1 - 35, 2019.


Received: 17 July 2018; Accepted: 27 February 2019; Published: 2019
First available in Project Euclid: 26 March 2019

zbMATH: 07055665
MathSciNet: MR3933206
Digital Object Identifier: 10.1214/19-EJP284

Primary: 60F05
Secondary: 60E15, 60G55, 60J27


Vol.24 • 2019
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