Electronic Communications in Probability

Transport cost estimates for random measures in dimension one

Martin Huesmann

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Abstract

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 $0<p<1$.

Article information

Source
Electron. Commun. Probab., Volume 21 (2016), paper no. 46, 10 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1464709607

Digital Object Identifier
doi:10.1214/16-ECP4590

Mathematical Reviews number (MathSciNet)
MR3510254

Zentralblatt MATH identifier
1345.60046

Subjects
Primary: 60G57: Random measures 60G55: Point processes
Secondary: 49Q20: Variational problems in a geometric measure-theoretic setting

Keywords
optimal transport random measures shift-coupling allocation extra head scheme

Rights
Creative Commons Attribution 4.0 International License.

Citation

Huesmann, Martin. Transport cost estimates for random measures in dimension one. Electron. Commun. Probab. 21 (2016), paper no. 46, 10 pp. doi:10.1214/16-ECP4590. https://projecteuclid.org/euclid.ecp/1464709607


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