Lévy metric

Template:Short description 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 be

$L(F,G):=\inf \{\epsilon >0|F(x-\epsilon )-\epsilon \le G(x)\le F(x+\epsilon )+\epsilon ,\mathrm{\forall }x\in ℝ\}.$

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 $\{F_n{\}}_{n=1}^{\mathrm{\infty }}$ 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

• Template:Springer

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