Compound Poisson process approximation. (Q1872302)

Point processes on the metric space \(\Gamma\) are considered. On the basis of the metric \(d_0\) on \(\Gamma\) the metric \(d_1\) on the space \({\mathcal X}\) of all finite subsets of \(\Gamma\) is defined. On the basis of the metric \(d_1\) the distance \(d_2\) between two probability measures on \({\mathcal X}\) is defined. To estimate the distance \(d_2(R,Q)\), where \(R\) is some measure on \({\mathcal X}\) and \(Q\) is the measure of a compound Poisson point process on \({\mathcal X}\), the authors use the Stein equation treating a kind of stochastic integral with respect to Poisson point process. The corresponding theorem is refined for some partial cases. It is applied to the problem of rare events in a stationary Markov chain, and to other problems.