Prokhorov's theorem

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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. However, it seems quite often being referred to as 'Helley's Theorem'(e.g., Billingsley 1985).

Contents 1 Statement of the theorem 2 Corollaries 3 Projective systems of measures 4 References

[edit] Statement 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).

1. 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)).
2. Conversely, if there exists an equivalent complete metric d₀ for (M, d) (so that (M, d₀) 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.

[edit] 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.

[edit] 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_ijπ_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 and Minlos' theorem.

[edit] References

• Billingsley, Patrick (1995). Probability and measure. New York, NY: John Wiley & Sons, Inc.. ISBN 0-471-00710-2. 
• Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc.. ISBN 0-471-19745-9. 
• Prokhorov, Yuri V. (1956). "Convergence of random processes and limit theorems in probability theory" (in English translation). Theory of Prob. and Appl. I 2: 157–214. doi:10.1137/1101016.