HOME

TheInfoList



OR:

In
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simil ...
Prokhorov's theorem relates tightness of measures to relative
compactness In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
(and hence weak convergence) in the space of
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more gener ...
s. It is credited to the Soviet mathematician
Yuri Vasilyevich Prokhorov Yuri Vasilyevich Prokhorov (russian: Ю́рий Васи́льевич Про́хоров; 15 December 1929 – 16 July 2013) was a Russian mathematician, active in the field of probability theory. He was a PhD student of Andrey Kolmogorov at th ...
, 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.


Statement

Let (S, \rho) be a separable metric space. Let \mathcal(S) denote the collection of all probability measures defined on S (with its Borel σ-algebra). Theorem. # A collection K\subset \mathcal(S) of probability measures is tight if and only if the closure of K is sequentially compact in the space \mathcal(S) equipped with the topology of weak convergence. # The space \mathcal(S) with the topology of weak convergence is metrizable. # Suppose that in addition, (S,\rho) is a complete metric space (so that (S,\rho) is a Polish space). There is a complete metric d_0 on \mathcal(S) equivalent to the topology of weak convergence; moreover, K\subset \mathcal(S) is tight if and only if the closure of K in (\mathcal(S),d_0) is compact.


Corollaries

For Euclidean spaces we have that: * If (\mu_n) is a tight sequence in \mathcal(\mathbb^m) (the collection of probability measures on m-dimensional Euclidean space), then there exist a subsequence (\mu_) and a probability measure \mu\in\mathcal(\mathbb^m) such that \mu_ converges weakly to \mu. * If (\mu_n) is a tight sequence in \mathcal(\mathbb^m) such that every weakly convergent subsequence (\mu_) has the same limit \mu\in\mathcal(\mathbb^m), then the sequence (\mu_n) converges weakly to \mu.


Extension

Prokhorov's theorem can be extended to consider
complex measure In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. Definition Formal ...
s or finite signed measures. Theorem: Suppose that (S,\rho) is a complete separable metric space and \Pi is a family of Borel complex measures on S. The following statements are equivalent: *\Pi is sequentially precompact; that is, every sequence \\subset\Pi has a weakly convergent subsequence. * \Pi is tight and uniformly bounded in
total variation norm In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function ''f'', defined on an interval ' ...
.


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

* * * *


References

* * * * {{Measure theory Compactness theorems Theorems in measure theory