Necessary and sufficient conditions for realizability of point processes
Kuna, T., Lebowitz, J. and Speer, E. (2011) Necessary and sufficient conditions for realizability of point processes. Annals of Applied Probability, 21 (4). pp. 1253-1281. ISSN 1050-5164
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To link to this article DOI: 10.1214/10-AAP703
We give necessary and sufficient conditions for a pair of (generali- zed) functions 1(r1) and 2(r1, r2), ri 2X, to be the density and pair correlations of some point process in a topological space X, for ex- ample, Rd, Zd or a subset of these. This is an infinite-dimensional version of the classical “truncated moment” problem. Standard tech- niques apply in the case in which there can be only a bounded num- ber 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 re- quirement—the existence of a finite third order moment. We general- ize the latter conditions in two distinct ways when X is not compact.
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