Lévy metric

From Wikipedia, the free encyclopedia

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.

[edit] Definition

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

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

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 $L (F, G)$ .