## Abstract

We giüe necessary and sufficient conditions for a pair of (generalized) functions ρ_{1} (r_{1}) and ρ_{2} (r_{1}, r_{1}), r_{i} ∈ X, to be the density and pair correlations of some point process in a topological space X, for example, R^{d} , Z^{d} or a subset of these. This is an infinite-dimensional üersion of the classical "truncated moment" problem. Standard techniques apply in the case in which there can be only a bounded number of points in any compact subset of X. Without this restriction we obtain, for compact X, strengthened conditions which are necessary and sufficient for the existence of a process satisfying a further requirement-the existence of a finite third order moment. We generalize the latter conditions in two distinct ways when X is not compact.

Original language | English (US) |
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Pages (from-to) | 1253-1281 |

Number of pages | 29 |

Journal | Annals of Applied Probability |

Volume | 21 |

Issue number | 4 |

DOIs | |

State | Published - Aug 2011 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Point processes
- Realizability
- Truncated moment problem