Lévy metric explained

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 : R \to [ 0 , 1 ] be two cumulative distribution functions. Define the Lévy distance between them to beL(F, G) : = inf \{ \varepsilon > 0 | F(x - \varepsilon) - \varepsilon \leq G(x) \leq F(x + \varepsilon) + \varepsilon, \forall x \in 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} \} ^{inf ty} _{n =1} 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