The Annals of Applied Probability

Necessary and sufficient conditions for realizability of point processes

Tobias Kuna, Joel L. Lebowitz, and Eugene R. Speer

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We give necessary and sufficient conditions for a pair of (generalized) functions ρ1(r1) and ρ2(r1, r2), riX, to be the density and pair correlations of some point process in a topological space X, for example, ℝd, ℤd or a subset of these. This is an infinite-dimensional version 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.

Article information

Ann. Appl. Probab., Volume 21, Number 4 (2011), 1253-1281.

First available in Project Euclid: 8 August 2011

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Zentralblatt MATH identifier

Primary: 60G55: Point processes 44A60: Moment problems
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs

Realizability point processes truncated moment problem


Kuna, Tobias; Lebowitz, Joel L.; Speer, Eugene R. Necessary and sufficient conditions for realizability of point processes. Ann. Appl. Probab. 21 (2011), no. 4, 1253--1281. doi:10.1214/10-AAP703.

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