Lévy distance

In stochastics, the Lévy distance , also called the Lévy metric , is a measure of the correspondence between two distribution functions . It is named after Paul Lévy and is a special case of the Prokhorov metric .

definition

Denote the set of all distribution functions (in terms of stochastics) . For two one defines ${\displaystyle {\mathcal {V}}_{1}}$ ${\displaystyle F,G\in {\mathcal {V}}_{1}}$

${\displaystyle d_{L}(F,G)=\inf\{\epsilon >0:F(x-\epsilon )-\epsilon \leq G(x)\leq F(x+\epsilon )+\epsilon {\text{ für alle }}x\in \mathbb {R} \}}$.

properties

• ${\displaystyle ({\mathcal {V}}_{1},d_{L})}$is a separable , complete metric space .
• The sequence of distribution functions converges weakly to a distribution function if and only if is. Thus metrisiert the Lévy metric the weak convergence of distribution functions. ${\displaystyle (F_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle \lim _{n\to \infty }d_{L}(F_{n},F)=0}$

Web links

• VM Zolotarev: Lévy metric . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

literature

• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .