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
 * $$A^{\varepsilon} := \{ p \in M | \exists q \in A, d(p, q) < \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p).$$

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

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

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

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

 * 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 $$\left( \mathcal{P} (M), \pi \right)$$ is separable if and only if $$(M, d)$$ is separable.
 * If $$\left( \mathcal{P} (M), \pi \right)$$ 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 $$\left( \mathcal{P} (M), \pi \right)$$ 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.