Teorema de Prokhorov

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.

Conteúdo 1 Declaração 2 Corolários 3 Extension 4 Comentários 5 Veja também 6 References Statement Let {estilo de exibição (S,rho )} be a separable metric space. Deixar {estilo de exibição {matemática {P}}(S)} denote the collection of all probability measures defined on {estilo de exibição S} (with its Borel σ-algebra).

Teorema.

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

Teorema: Suponha que {estilo de exibição (S,rho )} is a complete separable metric space and {displaystyle Pi } is a family of Borel complex measures on {estilo de exibição S} . The following statements are equivalent: {displaystyle Pi } is sequentially precompact; isso é, every sequence {estilo de exibição {dentro _{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. No entanto, 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). Convergência de Medidas de Probabilidade. Nova york, Nova Iorque: John Wiley & Sons, Inc. ISBN 0-471-19745-9. Bogachev, Vladimir (2006). Measure Theory Vol 1 e 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, Ricardo. M. (1989). Real analysis and Probability. Chapman & Hall. ISBN 0-412-05161-3. Categorias: Theorems in measure theoryCompactness theorems