Lévy–Prokhorov metric
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In mathematics, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
(i.e., a definition of distance) on the collection of probability measures on a given
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
. 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 In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy. Defi ...
.


Definition

Let (M, d) be a
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
with its
Borel sigma algebra In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are named ...
\mathcal (M). Let \mathcal (M) denote the collection of all probability measures on the
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. Definition Consider a set X and a σ-algebra \mathcal A on X. Then the ...
(M, \mathcal (M)). For a subset A \subseteq M, define the ε-neighborhood of A by :A^ := \ = \bigcup_ B_ (p). where B_ (p) is the
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defi ...
of radius \varepsilon centered at p. The Lévy–Prokhorov metric \pi : \mathcal (M)^ \to
open_ Open_or_OPEN_may_refer_to: _Music *__Open_(band),_Australian_pop/rock_band *__The_Open_(band),_English_indie_rock_band *__''Open''_(Blues_Image_album),_1969 *__''Open''_(Gotthard_album),_1999 *__''Open''_(Cowboy_Junkies_album),_2001 *__''Open''_(_...
_or_
open_ Open_or_OPEN_may_refer_to: _Music *__Open_(band),_Australian_pop/rock_band *__The_Open_(band),_English_indie_rock_band *__''Open''_(Blues_Image_album),_1969 *__''Open''_(Gotthard_album),_1999 *__''Open''_(Cowboy_Junkies_album),_2001 *__''Open''_(_...
_or_closed_set">closed_A;_either_inequality_implies_the_other,_and_(\bar)^\varepsilon_=_A^\varepsilon,_but_restricting_to_open_sets_may_change_the_metric_so_defined_(if_M_is_not_ open_ Open_or_OPEN_may_refer_to: _Music *__Open_(band),_Australian_pop/rock_band *__The_Open_(band),_English_indie_rock_band *__''Open''_(Blues_Image_album),_1969 *__''Open''_(Gotthard_album),_1999 *__''Open''_(Cowboy_Junkies_album),_2001 *__''Open''_(_...
_or_closed_set">closed_A;_either_inequality_implies_the_other,_and_(\bar)^\varepsilon_=_A^\varepsilon,_but_restricting_to_open_sets_may_change_the_metric_so_defined_(if_M_is_not_Polish_space">Polish_ Polish__may_refer_to: *_Anything_from_or_related_to_Poland,_a_country_in_Europe *_Polish_language *_Poles_ Poles,,_;_singular_masculine:_''Polak'',_singular_feminine:_''Polka''_or_Polish_people,_are_a__West_Slavic_nation_and__ethnic_group,_w_...
).


_Properties

*_If_(M,_d)_is_separable_space.html" ;"title="Polish_space.html" "title="closed_set.html" ;"title="open_set.html" "title=", + \infty) is defined by setting the distance between two probability measures \mu and \nu to be :\pi (\mu, \nu) := \inf \left\. For probability measures clearly \pi (\mu, \nu) \le 1. Some authors omit one of the two inequalities or choose only open set">open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' ( ...
 or closed set">closed A; either inequality implies the other, and (\bar)^\varepsilon = A^\varepsilon, but restricting to open sets may change the metric so defined (if M is not Polish_space">Polish Polish may refer to: * Anything from or related to Poland, a country in Europe * Polish language * Poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, w ...
).


Properties

* If (M, d) is separable space">separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to
weak convergence of measures In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing ...
. Thus, \pi is a Metrization theorem, metrization of the topology of weak convergence on \mathcal (M). * The metric space \left( \mathcal (M), \pi \right) is separable
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
(M, d) is separable. * If \left( \mathcal (M), \pi \right) is complete then (M, d) is complete. If all the measures in \mathcal (M) have separable support, then the converse implication also holds: if (M, d) is complete then \left( \mathcal (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 \subseteq \mathcal (M) is
relatively compact In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sin ...
if and only if its \pi-closure is \pi-compact. * If (M,d) is separable, then \pi (\mu , \nu ) = \inf \ , where \alpha (X,Y) = \inf\ 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 In mathematics, the Leonid Vaseršteĭn, Wasserstein distance or Leonid Kantorovich, Kantorovich–Gennadii Rubinstein, Rubinstein metric is a metric (mathematics), distance function defined between Probability distribution, probability distributi ...
with p\geq 1 and \mu, \nu have finite pth moment.


See also

*
Lévy metric In mathematics, the Lévy metric is a metric on the space of cumulative distribution functions of one-dimensional random variables. It is a special case of the Lévy–Prokhorov metric, and is named after the French mathematician Paul Lévy. Defi ...
* Prokhorov's theorem *
Tightness of measures In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity". Definitions Let (X, T) be a Hausdorff space, and let \Sigma be a σ-algebra on X that contai ...
*
weak convergence of measures In mathematics, more specifically measure theory, there are various notions of the convergence of measures. For an intuitive general sense of what is meant by ''convergence of measures'', consider a sequence of measures μ''n'' on a space, sharing ...
*
Wasserstein metric In mathematics, the Leonid Vaseršteĭn, Wasserstein distance or Leonid Kantorovich, Kantorovich–Gennadii Rubinstein, Rubinstein metric is a metric (mathematics), distance function defined between Probability distribution, probability distributi ...
* Radon distance * Total variation distance of probability measures


Notes


References

* * * * {{DEFAULTSORT:Levy-Prokhorov metric Measure theory Metric geometry Probability theory Paul Lévy (mathematician)