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 Lévy and the Soviet mathematician Yuri Vasilyevich 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
 * $$\pi (\mu, \nu) := \inf \left\{ \varepsilon > 0 ~|~ \mu(A) \leq \nu (A^{\varepsilon}) + \varepsilon \ \text{and} \ \nu (A) \leq \mu (A^{\varepsilon}) + \varepsilon \ \text{for all} \ A \in \mathcal{B}(M) \right\}.$$

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

Some authors omit one of the two inequalities or choose only open or closed $$A$$; either inequality implies the other, and $$(\bar{A})^\varepsilon = A^\varepsilon$$, but restricting to open sets may change the metric so defined (if $$M$$ is not Polish).

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 on $$\mathcal{P} (M)$$.
 * 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. In particular, this is the case if $$(M, d)$$ is separable.
 * 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.
 * If $$(M,d)$$ is separable, then $$\pi (\mu, \nu ) = \inf \{ \alpha (X,Y) : \text{Law}(X) = \mu , \text{Law}(Y) = \nu \} $$, where $$\alpha (X,Y) = \inf\{ \varepsilon > 0 : \mathbb{P} ( d( X ,Y ) > \varepsilon ) \leq \varepsilon \}$$ is the Ky Fan metric.

Relation to other distances
Let $$(M,d)$$ be separable. Then
 * $$ \pi (\mu, \nu ) \leq \delta (\mu , \nu) $$, where $$\delta (\mu,\nu)$$ is the total variation distance of probability measures
 * $$ \pi (\mu, \nu)^2 \leq W_p (\mu, \nu)^p$$, where $$W_p$$ is the Wasserstein metric with $$p\geq 1$$ and $$\mu, \nu$$ have finite $$p$$th moment.