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Gibbs point process approximation: total variation bounds using Stein’s method. (English) Zbl 1322.60060

Point processes (fields) on a compact metric space \(\mathcal X\) are considered. Such a point process is determined by a probability measure on the set \(\mathcal N\) of all finite subsets of this space. This probability measure depends on their family of conditional probabilities. A class of Gibbs processes is defined in terms of a partial ordering on the set \(\mathcal N\). The best known Gibbs process is constructed on the base of pairwise interaction of points from its realizations. This interaction is determined by some non-negative function \(\varphi\) on \(\mathcal X \times \mathcal X\). The Gibbs process is said to be a hard core process if there exists some \(a > 0\) such that \(\varphi (x_1 , x_2 ) = 1\) when \(\text{dist}(x_1 , x_2 ) \leq a\), and \(\varphi(x_1 , x_2 ) = 0\) when \(\text{dist}(x_1 , x_2 ) > a\). Another example of a Gibbs process is called Lennard-Jones process. It has an exponentially decreasing function \(\varphi\) as a function of distance between its arguments. A distance on \(\mathcal N\) and the corresponding distance on the set of measures on \(\mathcal N\) are defined. The distance between two Gibbs distributions is proved to be estimated from above in terms of a distance between their conditional intensities. These estimates permit to determine an approximation of general type Gibbs processes with the help of more investigated classes (for example, hard core Gibbs processes, or Lennard-Jones processes). The obtained results are applied for the solution of the coupling problem for two processes of birth and death. The main method of investigation used in the paper is a method of Stein with the proofs are being reduced to the solution of some integral-differential equation.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60G60 Random fields
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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