Loading AI tools
Relates tightness of measures to relative compactness in the space of probability measures From Wikipedia, the free encyclopedia
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.
Let be a separable metric space. Let denote the collection of all probability measures defined on (with its Borel σ-algebra).
Theorem.
For Euclidean spaces we have that:
Prokhorov's theorem can be extended to consider complex measures or finite signed measures.
Theorem: Suppose that is a complete separable metric space and is a family of Borel complex measures on . The following statements are equivalent:
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.
Seamless Wikipedia browsing. On steroids.
Every time you click a link to Wikipedia, Wiktionary or Wikiquote in your browser's search results, it will show the modern Wikiwand interface.
Wikiwand extension is a five stars, simple, with minimum permission required to keep your browsing private, safe and transparent.