Il teorema di 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.

Contenuti 1 Dichiarazione 2 Corollari 3 Extension 4 Commenti 5 Guarda anche 6 Riferimenti Dichiarazione Let {stile di visualizzazione (S,rho )} be a separable metric space. Permettere {stile di visualizzazione {matematico {P}}(S)} denote the collection of all probability measures defined on {stile di visualizzazione S} (with its Borel σ-algebra).

Teorema.

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

Teorema: Supporre che {stile di visualizzazione (S,rho )} is a complete separable metric space and {displaystyle Pi } is a family of Borel complex measures on {stile di visualizzazione S} . The following statements are equivalent: {displaystyle Pi } is sequentially precompact; questo è, every sequence {stile di visualizzazione {in _{n}}subset Pi } ha una sottosuccessione debolmente convergente. {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. Tuttavia, 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, Patrizio (1999). Convergenza delle misure di probabilità. New York, New York: 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, Richard. M. (1989). Real analysis and Probability. Chapman & Hall. ISBN 0-412-05161-3. Categorie: Theorems in measure theoryCompactness theorems