Prokhorov's theorem

Let ${\displaystyle (S,\rho )}$ be a separable metric space. Let ${\displaystyle {\mathcal {P}}(S)}$ denote the collection of all probability measures defined on ${\displaystyle S}$ (with its Borel σ-algebra).

Theorem.

1. A collection ${\displaystyle K\subset {\mathcal {P}}(S)}$ of probability measures is tight if and only if the closure of ${\displaystyle K}$ is sequentially compact in the space ${\displaystyle {\mathcal {P}}(S)}$ equipped with the topology of weak convergence.
2. The space ${\displaystyle {\mathcal {P}}(S)}$ with the topology of weak convergence is metrizable.
3. Suppose that in addition, ${\displaystyle (S,\rho )}$ is a complete metric space (so that ${\displaystyle (S,\rho )}$ is a Polish space). There is a complete metric ${\displaystyle d_{0}}$ on ${\displaystyle {\mathcal {P}}(S)}$ equivalent to the topology of weak convergence; moreover, ${\displaystyle K\subset {\mathcal {P}}(S)}$ is tight if and only if the closure of ${\displaystyle K}$ in ${\displaystyle ({\mathcal {P}}(S),d_{0})}$ is compact.

For Euclidean spaces we have that:

• If ${\displaystyle (\mu _{n})}$ is a tight sequence in ${\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}$ (the collection of probability measures on ${\displaystyle m}$-dimensional Euclidean space), then there exist a subsequence ${\displaystyle (\mu _{n_{k}})}$ and a probability measure ${\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}$ such that ${\displaystyle \mu _{n_{k}}}$ converges weakly to ${\displaystyle \mu }$.
• If ${\displaystyle (\mu _{n})}$ is a tight sequence in ${\displaystyle {\mathcal {P}}(\mathbb {R} ^{m})}$ such that every weakly convergent subsequence ${\displaystyle (\mu _{n_{k}})}$ has the same limit ${\displaystyle \mu \in {\mathcal {P}}(\mathbb {R} ^{m})}$, then the sequence ${\displaystyle (\mu _{n})}$ converges weakly to ${\displaystyle \mu }$.

Prokhorov's theorem can be extended to consider complex measures or finite signed measures.

Theorem: Suppose that ${\displaystyle (S,\rho )}$ is a complete separable metric space and ${\displaystyle \Pi }$ is a family of Borel complex measures on ${\displaystyle S}$. The following statements are equivalent:

• ${\displaystyle \Pi }$ is sequentially precompact; that is, every sequence ${\displaystyle \{\mu _{n}\}\subset \Pi }$ has a weakly convergent subsequence.
• ${\displaystyle \Pi }$ is tight and uniformly bounded in total variation norm.

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.

• Lévy–Prokhorov metric
• Sazonov's theorem
• Tightness of measures – Concept in measure theory
• Weak convergence of measures

• Billingsley, Patrick (1999). Convergence of Probability Measures. New York, NY: John Wiley & Sons, Inc. ISBN 0-471-19745-9.
• Bogachev, Vladimir (2006). Measure Theory Vol 1 and 2. Springer. ISBN 978-3-540-34513-8.
• Prokhorov, Yuri V. (1956). "Convergence of random processes and limit theorems in probability theory". Theory of Probability & Its Applications. 1 (2): 157–214. doi:10.1137/1101016.
• Dudley, Richard. M. (1989). Real analysis and Probability. Chapman & Hall. ISBN 0-412-05161-3.