Prokhorov's theorem


Prokhorov's theorem

In mathematics, Prokhorov's theorem is a theorem of measure theory that relates tightness of measures to weak compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilevich Prokhorov; it is also referred to as Helly's theorem (e.g., Billingsley 1985).

tatement of the theorem

Let ("M", "d") be a separable metric space, and let "P"("M") denote the collection of all probability measures defined on "M" (with its Borel σ-algebra).
# If a subset "K" of "P"("M") is a tight collection of probability measures, then "K" is relatively compact in "P"("M") with its topology of weak convergence (i.e., every sequence of measures in "K" has a subsequence that weakly converges to some measure in the (weak convergence)-closure of "K" in "P"("M")).
# Conversely, if there exists an equivalent complete metric "d"0 for ("M", "d") (so that ("M", "d"0) is a Polish space), then every relatively compact subset "K" of "P"("M") is also tight.

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.

Corollaries

If ("μ""n") is a tight sequence in "P"(R"k") (the collection of probability measures on "k"-dimensional Euclidean space), then there exists a subsequence ("μ""n"("i")) and probability measure "μ" in "P"(R"k") such that ("μ""n"("i")) converges weakly to "μ".

If ("μ""n") is a tight sequence in "P"(R"k"), and every subsequence of ("μ""n") that converges weakly at all converges weakly to the same probability measure "μ" in "P"(R"k"), then the full sequence ("μ""n") converges weakly to "μ".

Projective systems of measures

One version of Prokhorov's gives conditions for a projective systems of Radon probability measures to give a Radon measure as follows.

Suppose that "X" is a space with compatible maps to a projective system of spaces with Radon probability measures. This means that there is some ordered set "I" and that there is a Hausdorff space "X""i" with a Radon probability measure "&mu;""i" for each "i" in "I". Also for each "i" < "j" there is a map "&pi;""ij" from "X""i" to "X""j" taking "&mu;""i" to "&mu;""j". Finally "X" has maps "&pi;""i" to "X""i" such that "&pi;""i" = "&pi;""ij""&pi;""j".

In order that "X" has a Radon measure "&mu;" such that "&pi;""i"("&mu;") = "&mu;""i" for all "i" it is necessary and sufficient that the following "tightness" condition holds:
*For each "&epsilon;" &gt; 0 there is a compact subset "K" of "X" with "&mu;""i"("&pi;""i"("K")) &ge; 1 − &epsilon; for all "i". (The key point is that "K" does not depend on "i".)Moreover if the maps "&pi;""i" separate the points of "X" then "&mu;" is unique.

This version of Prokhorov's theorem is used to prove Sazonov's theorem and Minlos' theorem.

References

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