Lévy-Prokhorov metric

In mathematics, the Lévy-Prokhorov metric (sometimes known just as the Prokhorov metric) is a metric (i.e. a definition of distance) on the collection of probability measures on a given metric space. It is named after the French mathematician Paul Pierre Lévy and the Soviet mathematician Yuri Vasilevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.

Definition

Let $(M,\; d)$ be a metric space with its Borel sigma algebra $mathcal\{B\}\; (M)$. Let $mathcal\{P\}\; (M)$ denote the collection of all probability measures on the measurable space $(M,\; mathcal\{B\}\; (M))$.

For a subset $A\; subseteq\; M$, define the ε-neighborhood of $A$ by:

where $B\_\{varepsilon\}\; (p)$ is the open ball of radius $varepsilon$ centered at $p$.

The Lévy-Prokhorov metric $pi\; :\; mathcal\{P\}\; (M)^\{2\}\; o\; [0,\; +\; infty)$ is defined by setting the distance between two probability measures $mu$ and $u$ to be:$pi\; (mu,\; u)\; :=\; inf\; \{\; varepsilon\; >\; 0\; |\; mu\; (A)\; leq\; u\; (A^\{varepsilon\})\; +\; varepsilon\; mathrm\{,and,\}\; u\; (A)\; leq\; mu\; (A^\{varepsilon\})\; +\; varepsilon\; mathrm\{,for,all,\}\; A\; in\; mathcal\{B\}\; (M)\; \}.$

For probability measures clearly $pi\; (mu,\; u)\; leq\; 1$.

Some authors omit one of the two inequalities or choose only open or closed $A$; either inequality implies the other, but restricting to open/closed sets changes the metric so defined.

Properties

* If $(M,\; d)$ is separable, convergence of measures in the Lévy-Prokhorov metric is equivalent to weak convergence of measures. Thus, $pi$ is a metrization of the topology of weak convergence.
* The metric space $left(\; mathcal\{P\}\; (M),\; pi\; ight)$ is separable if and only if $(M,\; d)$ is separable.
* If $left(\; mathcal\{P\}\; (M),\; pi\; ight)$ is complete then $(M,\; d)$ is complete. If all the measures in $mathcal\{P\}\; (M)$ have separable support, then the converse implication also holds: if $(M,\; d)$ is complete then $left(\; mathcal\{P\}\; (M),\; pi\; ight)$ is complete.
* If $(M,\; d)$ is separable and complete, a subset $mathcal\{K\}\; subseteq\; mathcal\{P\}\; (M)$ is relatively compact if and only if its $pi$-closure is $pi$-compact.

ee also

* Lévy metric
* Wasserstein metric

References

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