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.
The Lévy–Prokhorov metric is defined by setting the distance between two probability measures and to be
For probability measures clearly .
Some authors omit one of the two inequalities or choose only
open or
closed; either inequality implies the other, and , but restricting to open sets may change the metric so defined (if is not
Polish).
If is
complete then is complete. If all the measures in have separable
support, then the converse implication also holds: if is complete then is complete. In particular, this is the case if is separable.
If is separable and complete, a subset is
relatively compact if and only if its -closure is -compact.
^Gibbs, Alison L.; Su, Francis Edward: On Choosing and Bounding Probability Metrics, International Statistical Review / Revue Internationale de Statistique, Vol 70 (3), pp. 419-435, Lecture Notes in Math., 2002.
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.
The Lévy–Prokhorov metric is defined by setting the distance between two probability measures and to be
For probability measures clearly .
Some authors omit one of the two inequalities or choose only
open or
closed; either inequality implies the other, and , but restricting to open sets may change the metric so defined (if is not
Polish).
If is
complete then is complete. If all the measures in have separable
support, then the converse implication also holds: if is complete then is complete. In particular, this is the case if is separable.
If is separable and complete, a subset is
relatively compact if and only if its -closure is -compact.
^Gibbs, Alison L.; Su, Francis Edward: On Choosing and Bounding Probability Metrics, International Statistical Review / Revue Internationale de Statistique, Vol 70 (3), pp. 419-435, Lecture Notes in Math., 2002.