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 Pierre Lévy and the Soviet mathematician Yuri Vasilevich Prokhorov; Prokhorov introduced it in 1956 as a generalization of the earlier Lévy metric.

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[edit] 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 μ and ν 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.

[edit] Properties

  • Convergence of measures in the Lévy-Prokhorov metric is equivalent to weak convergence of measures. Thus, π 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 π-closure is π-compact.

[edit] See also

[edit] References

  • Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9.
  • Zolotarev, V.M., "Lévy–Prokhorov metric" SpringerLink Encyclopaedia of Mathematics (2001)
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