mathematics

Hsu–Robbins–Erdős theorem

Hsu–Robbins–Erdős theorem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Hsu–Robbins–Erdős theorem rather than just read about it. In short: In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} is a sequence of i.i.d. random variables with zero mean and finite variance and S n = X 1 + ⋯ + X n , {\displaystyle S_{n}=X_{1}+\cdots +X_{n},\,} then ∑ n ⩾ 1 P ( | S n | > ε n ) < ∞ {\displaystyle \sum \limits _{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty } for every ε > 0 {\di…

Key takeaways

Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős theorem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hsu–Robbins–Erdős theorem from memory before moving on to harder problems.

Reference excerpt

In the mathematical theory of probability, the Hsu–Robbins–Erdős theorem states that if X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} is a sequence of i.i.d. random variables with zero mean and finite variance and

S n = X 1 + ⋯ + X n , {\displaystyle S_{n}=X_{1}+\cdots +X_{n},\,}

then

∑ n ⩾ 1 P ( | S n | > ε n ) < ∞ {\displaystyle \sum \limits _{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty }

for every ε > 0 {\displaystyle \varepsilon >0} . The result was proved by Pao-Lu Hsu and Herbert Robbins in 1947. This is an interesting strengthening of the classical strong law of large numbers in the direction of the Borel–Cantelli lemma. The idea of such a result is probably due to Robbins, but the method of proof is vintage Hsu. Hsu and Robbins further conjectured in that the condition of finiteness of the variance of X {\displaystyle X} is also a necessary condition for ∑ n ⩾ 1 P ( | S n | > ε n ) < ∞ {\displaystyle \sum \limits _{n\geqslant 1}P(|S_{n}|>\varepsilon n)<\infty } to hold. Two years later, the famed mathematician Paul Erdős proved the conjecture. Since then, many authors extended this result in several directions.

References

Worked examples

Example 1 — a first encounter with Hsu–Robbins–Erdős theorem

Start with the simplest possible case. Write down what Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős theorem

In research: Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős 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: Hsu–Robbins–Erdős theorem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Probabilistic inequalities, Theorems in measure theory, so understanding it makes those chapters shorter.
In everyday life: Look for Hsu–Robbins–Erdős 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.

Ask Teacher Smith questions about this articleOpens your AI tutor with a question about “Hsu–Robbins–Erdős theorem” →

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Hsu–Robbins–Erdős theorem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős theorem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hsu–Robbins–Erdős theorem in simple terms?

Why does Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős 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 Hsu–Robbins–Erdős theorem.