Tranport estimates for random measures in dimension one

TL;DR

This paper identifies a precise threshold in one-dimensional space for when the transport cost between Lebesgue measure and invariant random measures is finite, based on moments and variance conditions.

Contribution

It establishes sharp criteria and thresholds for the finiteness of transport costs between Lebesgue measure and invariant random measures in one dimension.

Findings

Finite transport cost depends on the divergence of first central moments.

Finite $L^p$ cost for $0<p<1$ is characterized by variance conditions.

Sharp thresholds are identified for different cost regimes.

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 \emph{any} such random measure the $L^1$ cost are 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 $0<p<1$.