Prokhorov's theorem

Key takeaways

Prokhorov's theorem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Prokhorov's theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Prokhorov's theorem from memory before moving on to harder problems.

Reference excerpt

In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements.

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

If a collection K ⊂ P ( S ) {\displaystyle K\subset {\mathcal {P}}(S)} of probability measures is tight, then the closure of K {\displaystyle K} is sequentially compact in the space P ( S ) {\displaystyle {\mathcal {P}}(S)} equipped with the topology of weak convergence. The space P ( S ) {\displaystyle {\mathcal {P}}(S)} with the topology of weak convergence is metrizable. Suppose that in addition, ( S , ρ ) {\displaystyle (S,\rho )} is a complete metric space (so that ( S , ρ ) {\displaystyle (S,\rho )} is a Polish space). Then the converse of (1) also holds; hence K ⊂ P ( S ) {\displaystyle K\subset {\mathcal {P}}(S)} is tight if and only if its closure is sequentially compact. Moreover, there exists a complete metric d 0 {\displaystyle d_{0}} on P ( S ) {\displaystyle {\mathcal {P}}(S)} equivalent to the topology of weak convergence, and for this metric K ⊂ P ( S ) {\displaystyle K\subset {\mathcal {P}}(S)} is tight if and only if its closure in ( P ( S ) , d 0 ) {\displaystyle ({\mathcal {P}}(S),d_{0})} is compact.

Corollaries For Euclidean spaces we have that:

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

Extension Prokhorov's theorem can be extended to consider complex measures or finite signed measures. Theorem: Suppose that ( S , ρ ) {\displaystyle (S,\rho )} is a complete separable metric space and Π {\displaystyle \Pi } is a family of Borel complex measures on S {\displaystyle S} . The following statements are equivalent:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Prokhorov's theorem

Start with the simplest possible case. Write down what Prokhorov's theorem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Prokhorov's theorem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Prokhorov's theorem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Prokhorov's theorem

In research: Prokhorov's theorem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Prokhorov's theorem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Prokhorov's theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Compactness theorems, Theorems in measure theory, so understanding it makes those chapters shorter.
In everyday life: Look for Prokhorov's theorem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Prokhorov's theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Prokhorov's theorem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Prokhorov's theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Prokhorov's theorem in simple terms?

In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spac…

Why does Prokhorov's theorem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Prokhorov's theorem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Prokhorov's theorem.