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 Pierre Lévy.

Definition

Let $F,\; G :\; \backslash mathbb\{R\}\; \backslash to\; [0,\; +\; \backslash infty)$ be two cumulative distribution functions. Define the Lévy distance between them to be

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

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

• Lévy-Prokhorov metric
• Wasserstein metric

References

• V.M. Zolotarev (2001), "Lévy metric", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104

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