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A note on disjoint-occurrence inequalities for marked Poisson point processes. (English) Zbl 0860.60013
The standard BK inequality for product measures on $$\{0,1\}^n$$ is a basic tool in lattice percolation theory [see the author and H. Kesten, ibid. 22, 556-569 (1985; Zbl 0571.60019)] and states that the probability that two increasing events “occur disjointly” is smaller than or equal to the product of the individual probabilities. In the paper generalizations are discussed. For (marked) Poisson point processes (for a class of increasing events) a new proof of the analog of the BK inequality is given. In contrast to the other proofs no extra topological conditions on the events (e.g. weak-convergence arguments) are needed. Apart from some well-known properties of Poisson point processes the “direct” proof is self-contained.