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| In [[measure theory]] '''Prokhorov’s theorem''' relates [[tightness of measures]] to relative [[compactness]] (and hence [[Weak convergence of measures|weak convergence]]) in the space of [[probability measure]]s. It is credited to the [[Soviet Union|Soviet]] mathematician [[Yuri Vasilevich Prokhorov|Yuri Vasilyevich Prokhorov]], who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements.
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| ==Statement of the theorem==
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| Let <math>(S, \rho)</math> be a [[separable space|separable]] [[metric space]].
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| Let <math>\mathcal{P}(S)</math> denote the collection of all probability measures defined on <math>S</math> (with its [[Borel sigma algebra|Borel σ-algebra]]).
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| '''Theorem.'''
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| # A collection <math>K\subset \mathcal{P}(S)</math> of probability measures is [[Tightness of measures|tight]] if and only if the closure of <math>K</math> is [[sequentially compact]] in the space <math>\mathcal{P}(S)</math> equipped with the [[topology]] of [[Weak convergence of measures|weak convergence]].
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| # The space <math>\mathcal{P}(S)</math> with the topology of weak convergence is [[metrizable]].
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| # Suppose that in addition, <math>(S,\rho)</math> is a [[complete metric space|complete metric]] (so that <math>(S,\rho)</math> is a [[Polish space]]). There is a complete metric <math>d_0</math> on <math>\mathcal{P}(S)</math> equivalent to the topology of weak convergence; moreover, <math> K\subset \mathcal{P}(S)</math> is tight if and only if the [[Closure (topology)|closure]] of <math> K</math> in <math>(\mathcal{P}(S),d_0)</math> is compact.
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| ==Corollaries==
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| For Euclidean spaces we have that:
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| * If <math> (\mu_n)</math> is a tight [[sequence]] in <math>\mathcal{P}(\mathbb{R}^k)</math> (the collection of probability measures on <math>k</math>-dimensional [[Euclidean space]]), then there exist a [[subsequence]] <math>(\mu_{n_k})</math> and a probability measure <math>\mu\in\mathcal{P}(\mathbb{R}^k)</math> such that <math>\mu_{n_k}</math> converges weakly to <math>\mu</math>.
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| * If <math> (\mu_n)</math> is a tight sequence in <math>\mathcal{P}(\mathbb{R}^k)</math> such that every weakly convergent subsequence <math>(\mu_{n_k})</math> has the same limit <math>\mu\in\mathcal{P}(\mathbb{R}^k)</math>, then the sequence <math>(\mu_n)</math> converges weakly to <math>\mu</math>.
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| ==Extension==
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| Prokhorov's theorem can be extended to consider [[complex measure]]s or finite [[signed measure]]s.
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| '''Theorem:'''
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| Suppose that <math>(S,\rho)</math> is a complete separable metric space and <math>\Pi</math> is a family of Borel complex measures on <math>S</math>.The following statements are equivalent:
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| *<math>\Pi</math> is sequentially compact; that is, every sequence <math>\{\mu_n\}\subset\Pi</math> has a weakly convergent subsequence.
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| * <math>\Pi</math> is tight and uniformly bounded in [[Total_variation#Total_variation_in_measure_theory|total variation norm]].
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| == Comments ==
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| Since Prokhorov's theorem expresses tightness in terms of compactness, the [[Arzelà-Ascoli theorem]] is often used to substitute for compactness: in function spaces, this leads to a characterization of tightness in terms of the [[modulus of continuity]] or an appropriate analogue — see [[Classical Wiener space#Tightness in classical Wiener space|tightness in classical Wiener space]] and [[Càdlàg#Tightness in Skorokhod space|tightness in Skorokhod space]].
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| There are several deep and non-trivial extension's to Prokhorov's theorem. However, those results do not overshadow the importance and the relevance to applications of the original result.
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| ==References==
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| <references/>
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| * {{cite book | last=Billingsley | first=Patrick | title=Convergence of Probability Measures | publisher=John Wiley & Sons, Inc. | location=New York, NY | year=1999 | isbn=0-471-19745-9}}
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| * {{cite book | last=Bogachev| first=Vladimir | title=Measure Theory Vol 1 and 2| publisher=Springer | year=2006| isbn=978-3-540-34513-8}}
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| * {{cite journal| last=Prokhorov | first=Yuri V.| title=Convergence of random processes and limit theorems in probability theory | journal=Theory of Prob. And Appl. I | volume=2 | year=1956 | pages=157–214 | language=English translation| doi=10.1137/1101016| issue=2 }}
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| * {{cite book | last=Dudley| first=Richard. M. | title=Real analysis and Probability| publisher=Chapman & Hall | year=1989 | isbn=0-412-05161-3 }}
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| [[Category:Theorems in measure theory]]
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| [[Category:Compactness theorems]]
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Oscar is what my spouse loves to call me and I totally dig that name. His family members life in South Dakota but his wife desires them to transfer. He is truly fond of performing ceramics but he is struggling to discover time for it. I am a meter reader.
My web blog: kgmcscs.realmind.net