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 – certain metric on space of finite measuresPages displaying wikidata descriptions as a fallback
• Sazonov's theorem
• Tightness of measures – Concept in measure theory
• Weak convergence of measures – Mathematical conceptPages displaying short descriptions of redirect targets

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• 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.