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| In [[mathematics]], the '''Lévy–Prokhorov metric''' (sometimes known just as the '''Prokhorov metric''') is a [[metric (mathematics)|metric]] (i.e., a definition of distance) on the collection of [[probability measure]]s on a given [[metric space]]. It is named after the French mathematician [[Paul Lévy (mathematician)|Paul 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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| ==Definition==
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| Let <math>(M, d)</math> be a [[metric space]] with its [[Borel sigma algebra]] <math>\mathcal{B} (M)</math>. Let <math>\mathcal{P} (M)</math> denote the collection of all [[probability measure]]s on the [[measurable space]] <math>(M, \mathcal{B} (M))</math>.
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| For a [[subset]] <math>A \subseteq M</math>, define the [[epsilon-neighborhood|ε-neighborhood]] of <math>A</math> by
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| :<math>A^{\varepsilon} := \{ p \in M ~|~ \exists q \in A, \ d(p, q) < \varepsilon \} = \bigcup_{p \in A} B_{\varepsilon} (p).</math>
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| where <math>B_{\varepsilon} (p)</math> is the [[open ball]] of radius <math>\varepsilon</math> centered at <math>p</math>.
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| The '''Lévy–Prokhorov metric''' <math>\pi : \mathcal{P} (M)^{2} \to [0, + \infty)</math> is defined by setting the distance between two probability measures <math>\mu</math> and <math>\nu</math> to be
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| :<math>\pi (\mu, \nu) := \inf \left\{ \varepsilon > 0 ~|~ \mu(A) \leq \nu (A^{\varepsilon}) + \varepsilon \ \text{and} \ \nu (A) \leq \mu (A^{\varepsilon}) + \varepsilon \ \text{for all} \ A \in \mathcal{B}(M) \right\}.</math>
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| For probability measures clearly <math>\pi (\mu, \nu) \le 1</math>.
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| Some authors omit one of the two inequalities or choose only [[open set|open]] or [[closed set|closed]] <math>A</math>; either inequality implies the other, but restricting to open/closed sets changes the metric so defined.
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| ==Properties==
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| * If <math>(M, d)</math> is [[separable space|separable]], convergence of measures in the Lévy–Prokhorov metric is equivalent to [[weak convergence of measures]]. Thus, <math>\pi</math> is a [[Metrization theorem|metrization]] of the topology of weak convergence.
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| * The metric space <math>\left( \mathcal{P} (M), \pi \right)</math> is [[separable space|separable]] [[if and only if]] <math>(M, d)</math> is separable.
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| * If <math>\left( \mathcal{P} (M), \pi \right)</math> is [[complete space|complete]] then <math>(M, d)</math> is complete. If all the measures in <math>\mathcal{P} (M)</math> have separable [[support (measure theory)|support]], then the converse implication also holds: if <math>(M, d)</math> is complete then <math>\left( \mathcal{P} (M), \pi \right)</math> is complete.
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| * If <math>(M, d)</math> is separable and complete, a subset <math>\mathcal{K} \subseteq \mathcal{P} (M)</math> is [[relatively compact]] if and only if its <math>\pi</math>-closure is <math>\pi</math>-compact.
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| ==See also==
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| * [[Lévy metric]]
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| * [[Wasserstein metric]]
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| ==References==
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| * {{cite book | author=Billingsley, Patrick | title=Convergence of Probability Measures | publisher=John Wiley & Sons, Inc., New York | year=1999 | isbn=0-471-19745-9 | oclc=41238534}}
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| * {{springer|author=Zolotarev, V.M.|id=l/l058320|title=Lévy–Prokhorov metric}}
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| {{DEFAULTSORT:Levy-Prokhorov metric}}
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| [[Category:Measure theory]]
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| [[Category:Metric geometry]]
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| [[Category:Probability theory]]
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Irwin Butts is what my wife loves to contact me though I don't truly like being known as like that. The favorite pastime for my children and me is to perform baseball but I haven't produced a dime with it. He utilized to be unemployed but now he is a meter reader. Her family members life in Minnesota.
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