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

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

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).

A sequence of cumulative distribution functions $\{F_{n}\}_{n=1}^{\infty }$ weakly converges to another cumulative distribution function $F$ if and only if $L(F_{n},F)\to 0$.

See also

• Càdlàg
• Lévy–Prokhorov metric
• Wasserstein metric

References

• V.M. Zolotarev (2001) [1994], "Lévy metric", Encyclopedia of Mathematics, EMS Press