Prokhorov's theorem

From Wikipedia, the free encyclopedia

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.

[edit] Statement of the theorem

Let $(M, d)$ be a separable metric space, and let $\mathcal{P}(M)$ denote the collection of all probability measures defined on $M$ (with its Borel sigma algebra).

If $\mathcal{K} \subseteq \mathcal{P}(M)$ is a tight collection of probability measures, then $\mathcal{K}$ is relatively compact in $\mathcal{P}(M)$ with its topology of weak convergence.
Conversely, if there exists an equivalent complete metric $d'$ for $M$ (so $(M, d')$ is a Polish space), then every relatively compact $\mathcal{K} \subseteq \mathcal{P}(M)$ 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 $(\mu_{n}) \subseteq \mathcal{P} (\mathbb{R}^{k})$ is a tight sequence of probability measures, then there exists a subsequence $(\mu_{n_{i}})$ and probability measure $\mu \in \mathcal{P} (\mathbb{R}^{k})$ such that $\mu_{n_{i}}$ converges weakly to $μ$ as $i \to \infty$ .

If $(\mu_{n}) \subseteq \mathcal{P} (\mathbb{R}^{k})$ is tight, and every subsequence that converges weakly at all converges weakly to the same probability measure $\mu \in \mathcal{P} (\mathbb{R}^{k})$ , then the full sequence $(μ n)$ converges weakly to $μ$ .

[edit] References

Billingsley, Patrick (1999). Convergence of Probability Measures. John Wiley & Sons, Inc., New York. ISBN 0-471-19745-9.
Billingsley, Patrick (1995). Probability and measure. John Wiley & Sons, Inc., New York. ISBN 0-471-00710-2.
Yu.V. Prokhorov. Convergence of random processes and limit theorems in probability theory. Theory of Prob. and Appl. I, no. 2 (1956), 157-214 (English translation).

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Categories: Measure theory | Mathematical theorems

Prokhorov's theorem

From Wikipedia, the free encyclopedia

[edit] Statement of the theorem

[edit] Corollaries

[edit] References

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