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
be a
separable metric space.
Let
denote the collection of all probability measures defined on
(with its
Borel σ-algebra).
Theorem.
# A collection
of probability measures is
tight if and only if the closure of
is
sequentially compact in the space
equipped with the
topology of
weak convergence.
# The space
with the topology of weak convergence is
metrizable.
# Suppose that in addition,
is a
complete metric space (so that
is a
Polish space). There is a complete metric
on
equivalent to the topology of weak convergence; moreover,
is tight if and only if the
closure of
in
is compact.
Corollaries
For Euclidean spaces we have that:
* If
is a tight
sequence in
(the collection of probability measures on
-dimensional
Euclidean space), then there exist a
subsequence 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 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
is a complete separable metric space and
is a family of Borel complex measures on
. The following statements are equivalent:
*
is sequentially precompact; that is, every sequence
has a weakly convergent subsequence.
*
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
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References
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{{Measure theory
Compactness theorems
Theorems in measure theory