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The scaling limit of Poisson-driven order statistics with applications in geometric probability
Let be a Poisson point process of intensity on some state
space \Y and be a non-negative symmetric function on \Y^k for some
. Applying to all -tuples of distinct points of
generates a point process on the positive real-half axis. The scaling
limit of as tends to infinity is shown to be a Poisson point
process with explicitly known intensity measure. From this, a limit theorem for
the the -th smallest point of is concluded. This is strengthened by
providing a rate of convergence. The technical background includes Wiener-It\^o
chaos decompositions and the Malliavin calculus of variations on the Poisson
space as well as the Chen-Stein method for Poisson approximation. The general
result is accompanied by a number of examples from geometric probability and
stochastic geometry, such as Poisson -flats, Poisson random polytopes,
random geometric graphs and random simplices. They are obtained by combining
the general limit theorem with tools from convex and integral geometry
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