Lévy–Prokhorov metric

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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.


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).


See also