### Abstract

Consider a measure μλ = Σx ξx δx where the sum is over points x of a Poisson point process of intensity λ on a bounded region in d-space, and ξx is a functional determined by the Poisson points near to x, i.e. satisfying an exponential stabilization condition, along with a moments condition (examples include statistics for proximity graphs, germ-grain models and random sequential deposition models). A known general result says the μλ-measures (suitably scaled and centred) of disjoint sets in Rd are asymptotically independent normals as λ tends to infinity; here we give an O( λ-1/(2d + ε)) bound on the rate of convergence. We illustrate our result with an explicit multivariate central limit theorem for the nearest-neighbour graph on Poisson points on a finite collection of disjoint intervals.

Original language | English |
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Pages (from-to) | 293-326 |

Number of pages | 23 |

Journal | Journal of Statistical Theory and Practice |

Volume | 2 |

Issue number | 2 |

Publication status | Published - 2008 |

### Keywords

- multivariate normal approximation
- geometric probability
- stabilization
- central limit theorem
- Stein's method
- nearest-neighbour graph
- statistics

## Cite this

Penrose, M. D., & Wade, A. R. (2008). Multivariate normal approximation in geometric probability.

*Journal of Statistical Theory and Practice*,*2*(2), 293-326.