Prokhorov metric

The Prokhorov metric is a metric on the set of finite measures . It thus clearly assigns a distance to two dimensions and thus makes it possible to formulate concepts of convergence. It is a generalization of the Lévy metric for distribution functions in the sense of stochastics (named after Paul Lévy ) and is therefore sometimes also called the Lévy-Prokhorow metric . In particular, it metrizes the weak convergence of measures . It was named after Yuri Wassiljewitsch Prokhorov , who dealt with it in the mid-1950s. Due to the different transcriptions of his name, there are also different spellings for this metric.

definition

Let be a metric space , the Borel σ-algebra on and the set of finite measures on the measurement space . Furthermore denote ${\displaystyle (E,d)}$ ${\displaystyle {\mathcal {B}}}$ ${\displaystyle E}$ ${\displaystyle {\mathcal {M}}_{f}^{+}(E,{\mathcal {B}})}$ ${\displaystyle (E,{\mathcal {B}})}$

${\displaystyle B_{\epsilon }:=\{x\in E\,|\,d(x,B)<\epsilon \}}$

the environment of the crowd . Defined for two ${\displaystyle \epsilon }$ ${\displaystyle B}$ ${\displaystyle \mu ,\nu \in {\mathcal {M}}_{f}^{+}(E,{\mathcal {B}})}$

${\displaystyle d^{*}(\mu ,\nu ):=\inf\{\epsilon >0\,|\,\mu (B)\leq \nu (B_{\epsilon })+\epsilon {\text{ für alle }}B\in {\mathcal {B}}\}}$,

then is called

${\displaystyle d_{P}(\mu ,\nu ):=\max\{d^{*}(\mu ,\nu ),d^{*}(\nu ,\mu )\}}$

the Prokhorov metric on a set of finite measures . ${\displaystyle {\mathcal {M}}_{f}^{+}(E,{\mathcal {B}})}$

properties

The Prokhorov metric turns into a metric space. The properties of this space depend essentially on the properties of . For example is ${\displaystyle ({\mathcal {M}}_{f}^{+}(E,{\mathcal {B}}),d_{P})}$ ${\displaystyle (E,d)}$

• ${\displaystyle (E,d)}$a separable metric space if and only if there is a separable space. ${\displaystyle ({\mathcal {M}}_{f}^{+}(E,{\mathcal {B}}),d_{P})}$
• ${\displaystyle (E,d)}$a Polish room if and only if there is a Polish room ${\displaystyle ({\mathcal {M}}_{f}^{+}(E,{\mathcal {B}}),d_{P})}$

In addition, the convergence on the Prokhorov metric implies the weak convergence of measures . If there is a separable metric space, the reverse is also true. Then convergence on the Prokhorov metric and weak convergence of measures are equivalent. The Prokhorov metric metrisiert then so on the topology of weak convergence . ${\displaystyle (E,d)}$ ${\displaystyle {\mathcal {M}}_{f}^{+}(E,{\mathcal {B}})}$

Special cases

If probability measures are and is a separable metric space, then and thus holds ${\displaystyle \mu ,\nu }$ ${\displaystyle E}$ ${\displaystyle d^{*}(\mu ,\nu )=d^{*}(\nu ,\mu )}$

${\displaystyle d_{P}(\mu ,\nu )=d^{*}(\mu ,\nu )=d^{*}(\nu ,\mu )}$

Since is Polish, for finite measures on the real numbers the convergence with respect to the Prokhorow metric according to the Helly-Bray theorem is equivalent to the weak convergence of distribution functions . ${\displaystyle \mathbb {R} }$

Accordingly, for probability measures on the real numbers, the convergence with respect to the Prokhorov metric is equivalent to the weak convergence of the distribution functions (in the sense of stochastics) and thus also equivalent to the convergence with respect to the Lévy distance .

Web links

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

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , pp. 401-408 , doi : 10.1007 / 978-3-540-89728-6 .
• Achim Klenke : Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , p. 257-258 , doi : 10.1007 / 978-3-642-36018-3 .