Electronic Journal of Probability

Multivariate approximation in total variation using local dependence

A.D. Barbour and A. Xia

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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.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 27, 35 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60E15: Inequalities; stochastic orderings 60G55: Point processes 60J27: Continuous-time Markov processes on discrete state spaces

total variation approximation Stein’s method local dependence marked point process

Creative Commons Attribution 4.0 International License.


Barbour, A.D.; Xia, A. Multivariate approximation in total variation using local dependence. Electron. J. Probab. 24 (2019), paper no. 27, 35 pp. doi:10.1214/19-EJP284. https://projecteuclid.org/euclid.ejp/1553565778

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