Stein’s method for compound Poisson approximation: The local approach. (English) Zbl 0816.60021

The author considers compound Poisson approximation by Stein’s method. Exploiting existence of a natural dependence structure which is appropriate for example in stationary \(m\)-mixing [M. R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and related properties of random sequences and processes (1983; Zbl 0518.60021)] or local dependence [A. D. Barbour, L. H. Y. Chen and W.-L. Loh, Ann. Probab. 20, No. 4, 1843-1866 (1992; Zbl 0765.60015)], the author gives a bound on the total variation distance between sums of dependent 0-1 valued random variables and the Poisson distribution. An analogue of this result for compound Poisson approximation is given and later it is applied to a reliability problem involving the number of isolated vertices in the rectangular lattice on the torus.


60F05 Central limit and other weak theorems
05C90 Applications of graph theory
60C05 Combinatorial probability
90B25 Reliability, availability, maintenance, inspection in operations research
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