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Transport cost estimates for random measures in dimension one. (English) Zbl 1345.60046

Summary: We show that there is a sharp threshold in dimension one for the transport cost between the Lebesgue measure \(\lambda \) and an invariant random measure \(\mu \) of unit intensity to be finite. We show that for any such random measure the \(L^1\) cost is infinite provided that the first central moments \(\mathbb{E} [|n-\mu ([0,n))|]\) diverge. Furthermore, we establish simple and sharp criteria, based on the variance of \(\mu ([0,n)]\), for the \(L^p\) cost to be finite for \(0<p<1\).

MSC:

60G57 Random measures
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
49Q20 Variational problems in a geometric measure-theoretic setting
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