Definitions

a prokhorov

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

  • 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 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.

See also

References

  • Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York.

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