Prokhorov's theorem

Prokhorov's theorem In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician 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.

Contents 1 Statement 2 Corollaries 3 Extension 4 Comments 5 See also 6 References Statement Let {displaystyle (S,rho )} be a separable metric space. Let {displaystyle {mathcal {P}}(S)} denote the collection of all probability measures defined on {displaystyle S} (with its Borel σ-algebra).

Theorem.

A collection {displaystyle Ksubset {mathcal {P}}(S)} of probability measures is tight if and only if the closure of {displaystyle K} is sequentially compact in the space {displaystyle {mathcal {P}}(S)} equipped with the topology of weak convergence. The space {displaystyle {mathcal {P}}(S)} with the topology of weak convergence is metrizable. Suppose that in addition, {displaystyle (S,rho )} is a complete metric space (so that {displaystyle (S,rho )} is a Polish space). There is a complete metric {displaystyle d_{0}} on {displaystyle {mathcal {P}}(S)} equivalent to the topology of weak convergence; moreover, {displaystyle Ksubset {mathcal {P}}(S)} is tight if and only if the closure of {displaystyle K} in {displaystyle ({mathcal {P}}(S),d_{0})} is compact. Corollaries For Euclidean spaces we have that: If {displaystyle (mu _{n})} is a tight sequence in {displaystyle {mathcal {P}}(mathbb {R} ^{m})} (the collection of probability measures on {displaystyle m} -dimensional Euclidean space), then there exist a subsequence {displaystyle (mu _{n_{k}})} and a probability measure {displaystyle mu in {mathcal {P}}(mathbb {R} ^{m})} such that {displaystyle mu _{n_{k}}} converges weakly to {displaystyle mu } . If {displaystyle (mu _{n})} is a tight sequence in {displaystyle {mathcal {P}}(mathbb {R} ^{m})} such that every weakly convergent subsequence {displaystyle (mu _{n_{k}})} has the same limit {displaystyle mu in {mathcal {P}}(mathbb {R} ^{m})} , then the sequence {displaystyle (mu _{n})} converges weakly to {displaystyle mu } . Extension Prokhorov's theorem can be extended to consider complex measures or finite signed measures.

Theorem: Suppose that {displaystyle (S,rho )} is a complete separable metric space and {displaystyle Pi } is a family of Borel complex measures on {displaystyle S} . The following statements are equivalent: {displaystyle Pi } is sequentially precompact; that is, every sequence {displaystyle {mu _{n}}subset Pi } has a weakly convergent subsequence. {displaystyle Pi } is tight and uniformly bounded in total variation norm. Comments 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 tightness in classical Wiener space and tightness in Skorokhod space.

There are several deep and non-trivial extensions to Prokhorov's theorem. However, those results do not overshadow the importance and the relevance to applications of the original result.

See also Lévy–Prokhorov metric Tightness of measures weak convergence of measures References Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9. Bogachev, Vladimir (2006). Measure Theory Vol 1 and 2. Springer. ISBN 978-3-540-34513-8. Prokhorov, Yuri V. (1956). "Convergence of random processes and limit theorems in probability theory". Theory of Probability & Its Applications. 1 (2): 157–214. doi:10.1137/1101016. Dudley, Richard. M. (1989). Real analysis and Probability. Chapman & Hall. ISBN 0-412-05161-3. Categories: Theorems in measure theoryCompactness theorems

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