Lévy metric

In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy.

Definition

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

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

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).

See also

• Càdlàg
• Lévy–Prokhorov metric
• Wasserstein metric

References

• V.M. Zolotarev (2001), "Lévy metric", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4