U-Statistics in Stochastic Geometry

  title={U-Statistics in Stochastic Geometry},
  author={Raphael Lacheze-Rey and Matthias Reitzner},
  journal={arXiv: Probability},
A U-statistic of order k with kernel \(f: \mathbb{X}^{k} \rightarrow \mathbb{R}^{d}\) over a Poisson process η is defined as $$\displaystyle{\sum _{(x_{1},\ldots,x_{k})}f(x_{1},\ldots,x_{k}),}$$ where the summation is over k-tuples of distinct points of η, under appropriate integrability assumptions on f. U-statistics play an important role in stochastic geometry since many interesting functionals can be written as U-statistics, like intrinsic volumes of intersection processes… 

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