In measure theory Prokhorov’s theorem relates tightness of measures

Tightness of measures

In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."-Definitions:...

 to weak compactness (and hence weak convergence) in the space of probability measure

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...

s. It is credited to the Soviet mathematician Yuri Vasilevich Prokhorov

Yuri Vasilevich Prokhorov

Yuri Vasilevich Prokhorov is a Russian mathematician, active in the field of probability theory. He was a PhD student of Andrey Nikolaevich Kolmogorov at the Moscow State University, where he obtained his PhD in 1949....

, 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 of the theorem

Let be a separable metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

.
Let denote the collection of all probability measures defined on (with its Borel σ-algebra).

Theorem.

1. A collection of probability measures is tight
   Tightness of measures
   In mathematics, tightness is a concept in measure theory. The intuitive idea is that a given collection of measures does not "escape to infinity."-Definitions:...
   
    if and only if is sequentially compact in the space equipped with the topology
   Topology
   Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
   
    of weak convergence.
2. The space with the topology of weak convergence is metrizable.
3. Suppose that in addition, is a complete metric (so that is a Polish space
   Polish space
   In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space; that is, a space homeomorphic to a complete metric space that has a countable dense subset. Polish spaces are so named because they were first extensively studied by Polish...
   
   ). There is a complete metric on equivalent to the topology of weak convergence; moreover, is tight if and only if the closure
   Closure (topology)
   In mathematics, the closure of a subset S in a topological space consists of all points in S plus the limit points of S. Intuitively, these are all the points that are "near" S. A point which is in the closure of S is a point of closure of S...
   
    of in is compact.

Corollaries

For Euclidean spaces we have that:

• If is a tight sequence
  Sequence
  In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
  
   in (the collection of probability measures on -dimensional Euclidean space
  Euclidean space
  In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
  
  ), then there exist a subsequence
  Subsequence
  In mathematics, a subsequence is a sequence that can be derived from another sequence by deleting some elements without changing the order of the remaining elements...
  
    and a probability measure such that converges weakly to .

• If is a tight sequence in such that every weakly convergent subsequence has the same limit , then the sequence converges weakly to .

Extension

Prokhorov's theorem can be extended to consider complex measures

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 is a complex number.-Definition:...

 or finite signed measures

Signed measure

In mathematics, signed measure is a generalization of the concept of measure by allowing it to have negative values. Some authors may call it a charge, by analogy with electric charge, which is a familiar distribution that takes on positive and negative values.-Definition:There are two slightly...

.

Theorem:
Suppose that is a complete separable metric space and is a family of Borel complex measures on .The following statements are equivalent:

• is sequentially compact; that is, every sequence has a weakly convergent subsequence
• is tight and uniformly bounded in total variation norm.

Comments

Since Prokhorov's theorem expresses tightness in terms of compactness, the Arzelà-Ascoli theorem

Arzelà-Ascoli theorem

In mathematics, the Arzelà–Ascoli theorem of functional analysis gives necessary and sufficient conditions to decide whether every subsequence of a given sequence of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition...

 is often used to substitute for compactness: in function spaces, this leads to a characterization of tightness in terms of the modulus of continuity

Modulus of continuity

In mathematical analysis, a modulus of continuity is a function\omega:[0,\infty]\to[0,\infty]used to measure quantitatively the uniform continuity of functions. So, a function f:I\to\R admits \omega as a modulus of continuity if and only if|f-f|\leq\omega,for all x and y in the domain of f...

 or an appropriate analogue — see tightness in classical Wiener space and tightness in Skorokhod space.

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