Lévy metric

Let ${\displaystyle F,G:\mathbb {R} \to [0,1]}$ be two cumulative distribution functions. Define the Lévy distance between them to be

${\displaystyle L(F,G):=\inf\{\varepsilon >0|F(x-\varepsilon )-\varepsilon \leq G(x)\leq F(x+\varepsilon )+\varepsilon \mathrm {\,for\,all\,} x\in \mathbb {R} \}.}$

Intuitively, if between the graphs of F and G one inscribes squares with sides parallel to the coordinate axes (at points of discontinuity of a graph vertical segments are added), then the side-length of the largest such square is equal to L(F, G).

• Càdlàg
• Lévy–Prokhorov metric
• Wasserstein metric

• V.M. Zolotarev (2001) [1994], "Lévy metric", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4