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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.

MSC:

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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