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The Stein-Chen method, point processes and compensators. (English) Zbl 0770.60052

There are two methods used for obtaining bounds on the total variation accuracy of approximation of the distribution of a sum \(N_ n\) of dependent indicators \(I_ 1,\dots,I_ n\) by a Poisson distribution. The first one [see D. Freedman, Ann. Probab. 2, 256-269 (1974; Zbl 0301.60021)] is to suppose that there is an associated filtration \(({\mathcal F}_ i)_ i\) and probabilities \(p_ i=P[I_ i=1/{\mathcal F}_{i-1}]\) are known for each \(i\). A typical result is [see T. C. Brown, ibid. 11, 726-744 (1983; Zbl 0551.60048)] \[ d_{TV}({\mathcal L}N_ n,\text{Poisson}_ \mu)\leq E\left|\sum_{j\leq n}p_ j- \mu\right|+E\left\{\sum_{j\leq n}p^ 2_ j\right\}, \tag{1} \] where \(d_{TV}\) denotes total variation distance. The second method was developed by L. H. Y. Chen [ibid. 3, 534-545 (1975; Zbl 0335.60016)]. Here, the dependence structures are such that any given indicator is only strongly dependent on a few others or such that the dependence between indicators is essentially symmetrical. The aim of this paper is to combine the two methods in such a way that the extra precision of the Stein-Chen approach can be achieved for proint processes, especially taking into account the fact that point processes are often defined by their compensators. There are two main results. The first is a generalization of Theorem 2.1 of A. D. Barbour and L. Holst [Adv. Appl. Probab. 21, No. 1, 74-90 (1989; Zbl 0673.60023)] to a point process \(N\) over a general carrier space. The second result, Theorem 3.7, presupposes a point process \(N\) on the line, and the bound consists of three terms. Two of these are those of (1), multiplied by factors which become small if \(\mu\) is large.
Reviewer: V.Oganyan (Erevan)

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60E15 Inequalities; stochastic orderings
60G44 Martingales with continuous parameter
60J75 Jump processes (MSC2010)
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