Azuma's Inequality, also called the Azuma-Hoeffding Inequality, is a concentration inequality bounding how far a martingale with bounded increments can stray from its starting value after a given number of steps, showing the probability of a large deviation falls off exponentially as the deviation grows. Named for Kazuoki Azuma and Wassily Hoeffding, it is a foundational tool of probability theory used throughout combinatorics and theoretical computer science to prove that quantities built from many small, weakly dependent random choices concentrate tightly around their expected value.
Facts
StatementGives a concentration result for the values of martingales that have bounded differences, bounding P(X_N - X_0 >= epsilon) by exp(-epsilon^2 / (2 sum of c_k^2)). 2 Classification
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Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
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Source Azuma's inequality (Wikipedia)
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Source Azuma's inequality (Wikipedia)
Sources
1. Wikipedia: Azuma's inequality
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
In probability theory, Azuma's inequality or the Azuma-Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales that have bounded differences.
2. Azuma's inequality (Wikipedia)
Introduction, sentence 1
gives a concentration result for the values of martingales that have bounded differences
In Branch: Probability and Statistics, Lead sentence
In probability theory, Azuma's inequality or the Azuma-Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) give
Proved By: Wassily Hoeffding, Lead paragraph
In probability theory, Azuma's inequality or the Azuma-Hoeffding inequality (named after Kazuoki Azuma and Wassily Hoeffding) gives a concentration result for the values of martingales
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