## 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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