## Poisson approximation in connection with clustering of random points.(English)Zbl 0941.60027

Let $$n$$ particles be independently and uniformly distributed in a rectangle $${\mathbf A}\subset\mathbb{R}$$. The number of $$k$$ subsets of particles which are actually covered by some translate of a given convex set $${\mathbf A}\subset{\mathbf C}$$ is denoted by $$W$$. The positions of such subsets constitute a point process on $${\mathbf A}$$. Each point of this process can be marked with the smallest necessary “size” of a set, of the same shape and orientation as $$C$$, which covers the particles determining the point. This paper considers Poisson (process) approximations of $$W$$ and of the above point process, by way of Stein’s method.

### MSC:

 60D05 Geometric probability and stochastic geometry 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60F17 Functional limit theorems; invariance principles

### Keywords:

Poisson (process) approximations; Stein’s method
Full Text:

### References:

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