Transport cost estimates for random measures in dimension one

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 $0<p<1$.