- Prokhorov's theorem
In

mathematics ,**Prokhorov's theorem**is atheorem ofmeasure theory that relatestightness of measures to weakcompactness (and hence weak convergence) in the space ofprobability measure s. It is credited to theSoviet 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" isrelatively compact in**"P**"("M") with itstopology 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 aPolish 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 themodulus of continuity or an appropriate analogue — see tightness in classical Wiener space and tightness in Skorokhod space.**Corollaries**If ("μ"

_{"n"}) is a tightsequence in**"P**"(**R**^{"k"}) (the collection of probability measures on "k"-dimensionalEuclidean space ), then there exists asubsequence ("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 "μ"_{"i"}for each "i" in "I". Also for each "i" < "j" there is a map "π"_{"ij"}from "X"_{"i"}to "X"_{"j"}taking "μ"_{"i"}to "μ"_{"j"}. Finally "X" has maps "π"_{"i"}to "X"_{"i"}such that "π"_{"i"}= "π"_{"ij"}"π"_{"j"}.In order that "X" has a Radon measure "μ" such that "π"

_{"i"}("μ") = "μ"_{"i"}for all "i" it is necessary and sufficient that the following "tightness" condition holds:

*For each "ε" > 0 there is a compact subset "K" of "X" with "μ"_{"i"}("π"_{"i"}("K")) ≥ 1 − ε for all "i". (The key point is that "K" does not depend on "i".)Moreover if the maps "π"_{"i"}separate the points of "X" then "μ" is unique.This version of Prokhorov's theorem is used to prove

Sazonov's theorem andMinlos' theorem .**References***

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