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Limit theorems for point processes under geometric constraints (and topological crackle). (English) Zbl 1373.60090

Let \(\{X_i: i \geq 1\}\) be a sequence of i.i.d. \(d\)-dimensional random variables with spherically symmetric density \(f\) , \(N_n\), \(n \geq 1\) be an independent Poisson random variable with mean \(n\), and for \(k \geq 1\), let \(N_n^{(k)}\) denote the number of \(k\)-tuples of the \(X_i\), \(i \leq N_n\) satisfying certain specified geometric constraints. Finally, let \(N^{(k)}\) be a \(k\)-dimensional Poisson random measure. First, the weak convergence of \(N_n^{(k)}\) to \(N^{(k)}\) is established. More detailed convergence results are given separately for the cases where the density \(f\) has heavy or light tail. The results are applied to obtain limiting results for sequences of partial maxima and sums, and for Betti numbers of the Čech complex that arise in topology. The case of \(k=1\) is closely linked to the results in classical extreme value theory.

MSC:

60G70 Extreme value theory; extremal stochastic processes
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05 Geometric probability and stochastic geometry
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