Open Access
2016 Transport cost estimates for random measures in dimension one
Martin Huesmann
Electron. Commun. Probab. 21: 1-10 (2016). DOI: 10.1214/16-ECP4590


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 $0<p<1$.


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Martin Huesmann. "Transport cost estimates for random measures in dimension one." Electron. Commun. Probab. 21 1 - 10, 2016.


Received: 28 September 2015; Accepted: 19 April 2016; Published: 2016
First available in Project Euclid: 31 May 2016

zbMATH: 1345.60046
MathSciNet: MR3510254
Digital Object Identifier: 10.1214/16-ECP4590

Primary: 60G55 , 60G57
Secondary: 49Q20

Keywords: Allocation , extra head scheme , Optimal transport , Random measures , Shift-coupling

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