Necessary and sufficient conditions for realizability of point processes

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

Research output: Contribution to journalArticlepeer-review

16 Scopus citations


We giüe necessary and sufficient conditions for a pair of (generalized) functions ρ1 (r1) and ρ2 (r1, r1), ri ∈ X, 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 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 languageEnglish (US)
Pages (from-to)1253-1281
Number of pages29
JournalAnnals of Applied Probability
Issue number4
StatePublished - Aug 2011

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Point processes
  • Realizability
  • Truncated moment problem


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