KS distance

The Kolmogorov-Smirnoff distributional distance between two distributions $P_X$ and $P_Y$ over the reals is

$${\rho }_{\text{KS}}(X,Y)=\underset{z\in \mathbb{R}}{\sup }|P_X(X\le z)-P_Y(Y\le z)|.$$

Note that $P_X$ and $P_Y$ need not have the same support for this definition to make sense. They could even be defined on different probability spaces.

The KS distance is an ideal metric of order 0. It is useful for proving central limit theorems and provides an error bound on the Wald interval. Berry-Esseen bounds are often stated in terms of the KS distance.

KS distance is useful for CLTs because, for a sequence of random variables $X_1,X_2,\dots$, if ${\rho }_{\text{KS}}(X_n\Vert X_{\infty })\to 0$ then $X_n\stackrel{d}{\to }{X}_{\infty }$. Moreover, the converse is true if $X_{\infty }$ is a continuous random variable.

The KS distance is obviously upper bounded by the total variation distance.