ideal metrics

An ideal metric $\rho$ of order $s$ is a distributional distance that is both regular and homogeneous of order $s$. That is, it satisfies $\rho (X+Z,Y+Z)\le \rho (X,Y)$ for any $Z$ independent of $X$ and $Y$ and

$$\rho (cX,cY)=|c{|}^s\rho (X,Y)\text{for all }c\in \mathbb{R}.$$

(See more in distributional distance - definitions). The KS distance is ideal of order 0, and the Wasserstein distance is ideal of order 1.

To see how they can be useful for proving central limit theorems, see quantitative CLT template with ideal metrics.

Do ideal metrics of an arbitrary order $s\ge 0$ always exist? This question was answered affirmatively by VM Zolotarev in 1975. It is nicely described by Senatov in his book Normal approximation: New Results, Methods, and Problems.

For any $s$, write $s=m+\alpha$ where $m\in \mathbb{Z}$ and $\alpha \in [0,1]$. Consider the set of functions ${\mathcal{F}}_s$ which have Holder exponent $\alpha$, i.e..,

$$|f^{(m)}(x)-f^{(m)}(y)|\le {\left|x-y\right|}^{\alpha }.$$

If we define the distance

$${\rho }_s(P,Q)=\underset{f\in {\mathcal{F}}_s}{\sup }\left\{\left|\int f(x)(P-Q)(dx)\right|\right\},$$

then this is an ideal metric of order $s$.