Lévy–Prokhorov Metric
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Lévy–Prokhorov metric (sometimes known just as the Prokhorov metric) is a
metric Metric or metrical may refer to: Measuring * 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 ...
(i.e., a definition of distance) on the collection of
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s on a given
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
. 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. Def ...
.


Definition

Let (M, d) be a
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
with its
Borel sigma algebra In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are ...
\mathcal (M). Let \mathcal (M) denote the collection of all
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
s 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. It captures and generalises intuitive notions such as length, area, an ...
(M, \mathcal (M)). For a
subset In mathematics, a Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they a ...
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 defin ...
of radius \varepsilon centered at p. The Lévy–Prokhorov metric \pi : \mathcal (M)^ \to
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or 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).


Properties

* If (M, d) is separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to separable space">separable, convergence of measures in the Lévy–Prokhorov metric is equivalent to metrization of the topology of weak convergence on \mathcal (M). * The metric space \left( \mathcal (M), \pi \right) is separable
separable space">separable if and only if (M, d) is separable. * If \left( \mathcal (M), \pi \right) is complete space">complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...
then (M, d) is complete. If all the measures in \mathcal (M) have separable support (measure theory)">support Support may refer to: Arts, entertainment, and media * Supporting character * Support (art), a solid surface upon which a painting is executed Business and finance * Support (technical analysis) * Child support * Customer support * Income Su ...
, 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 if and only if its \pi-closure is \pi-compact. * If (M,d) is separable space, 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 Wasserstein distance or Kantorovich– Rubinstein metric is a distance function defined between probability distributions on a given metric space M. It is named after Leonid Vaseršteĭn. Intuitively, if each distribution ...
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. Def ...
* 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 on a space, sharing a com ...
*
Wasserstein metric In mathematics, the Wasserstein distance or Kantorovich– Rubinstein metric is a distance function defined between probability distributions on a given metric space M. It is named after Leonid Vaseršteĭn. Intuitively, if each distribution ...
* Radon distance * Total variation distance of probability measures


Notes


References

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