mathematics

Lenglart's inequality

Lenglart's inequality 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 Lenglart's inequality rather than just read about it. In short: In the mathematical theory of probability, Lenglart's inequality was proved by Érik Lenglart in 1977. Later slight modifications are also called Lenglart's inequality.

Key takeaways

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

Reference excerpt

In the mathematical theory of probability, Lenglart's inequality was proved by Érik Lenglart in 1977. Later slight modifications are also called Lenglart's inequality.

Statement Let X be a non-negative right-continuous F t {\displaystyle {\mathcal {F}}_{t}} -adapted process and let G be a non-negative right-continuous non-decreasing predictable process such that E [ X ( τ ) ∣ F 0 ] ≤ E [ G ( τ ) ∣ F 0 ] < ∞ {\displaystyle \mathbb {E} [X(\tau )\mid {\mathcal {F}}_{0}]\leq \mathbb {E} [G(\tau )\mid {\mathcal {F}}_{0}]<\infty } for any bounded stopping time τ {\displaystyle \tau } . Then

References

Citations

General sources Geiss, Sarah; Scheutzow, Michael (2021). "Sharpness of Lenglart's domination inequality and a sharp monotone version". Electronic Communications in Probability. 26: 1–8. arXiv:2101.10884. doi:10.1214/21-ECP413. S2CID 231709277. Lenglart, Érik (1977). "Relation de domination entre deux processus". Annales de l'Institut Henri Poincaré B. 13 (2): 171−179. Mehri, Sima; Scheutzow, Michael (2021). "A stochastic Gronwall lemma and well-posedness of path-dependent SDEs driven by martingale noise". Latin American Journal of Probability and Mathematical Statistics. 18: 193−209. arXiv:1908.10646. doi:10.30757/ALEA.v18-09. S2CID 201660248. Ren, Yaofeng; Schen, Jing (2012). "A note on the domination inequalities and their applications". Statist. Probab. Lett. 82 (6): 1160−1168. doi:10.1016/j.spl.2012.03.002. Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion. Berlin: Springer. ISBN 3-540-64325-7.

Worked examples

Example 1 — a first encounter with Lenglart's inequality

Start with the simplest possible case. Write down what Lenglart's inequality 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 Lenglart's inequality 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 Lenglart's inequality 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 Lenglart's inequality

In research: Lenglart's inequality 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 Lenglart's inequality 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: Lenglart's inequality is common in secondary-school and first-year university syllabi. It links to neighbouring topics Probabilistic inequalities, Stochastic differential equations, so understanding it makes those chapters shorter.
In everyday life: Look for Lenglart's inequality 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 “Lenglart's inequality” →

Affiliate

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

How to study Lenglart's inequality in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lenglart's inequality 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 Lenglart's inequality out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lenglart's inequality in simple terms?

In the mathematical theory of probability, Lenglart's inequality was proved by Érik Lenglart in 1977. Later slight modifications are also called Lenglart's inequality.

Why does Lenglart's inequality 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 Lenglart's inequality?

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 Lenglart's inequality.