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 [math]\displaystyle{ F, G : \mathbb{R} \to [0, 1] }[/math] be two cumulative distribution functions. Define the Lévy distance between them to be
- [math]\displaystyle{ L(F, G) := \inf \{ \varepsilon \gt 0 | F(x - \varepsilon) - \varepsilon \leq G(x) \leq F(x + \varepsilon) + \varepsilon,\; \forall x \in \mathbb{R} \}. }[/math]
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 [math]\displaystyle{ \{F_n \}_{n=1}^\infty }[/math] weakly converges to another cumulative distribution function [math]\displaystyle{ F }[/math] if and only if [math]\displaystyle{ L(F_n,F) \to 0 }[/math].
See also
References
- Hazewinkel, Michiel, ed. (2001), "Lévy metric", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=l/l058310
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