CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Symmetric bounded-increment Azuma bound

Statement

Assume AC. If $(M_{k}, F_{k})_{k = 0}^{n}$ is a martingale and $∣ M_{k} - M_{k - 1} ∣ \leq c_{k}$ almost surely for deterministic $c_{k} \geq 0$ , then for every $t > 0$ , $P (∣ M_{n} - M_{0} ∣ \geq t) \leq 2 \exp ⁣ (- \frac{t^{2}}{2 \sum_{k = 1}^{n} c_{k}^{2}}),$ again interpreting the right exponential as $0$ when every $c_{k} = 0$ .

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Azuma-Hoeffding inequality supplies both one-sided bounds with predictable endpoints.

[F2]

The finite union bound combines the upper and lower deviations.

[F3]

The Axiom of Choice is the exact inherited conditional-expectation dependence from F1.

Proof

1.1F1

Apply F1 with $A_{k} = - c_{k}$ and $B_{k} = c_{k}$ . Their width is $2 c_{k}$ , so each one-sided probability is at most $\exp ⁣ (- \frac{2 t^{2}}{\sum_{k} (2 c_{k})^{2}}) = \exp ⁣ (- \frac{t^{2}}{2 \sum_{k} c_{k}^{2}}) .$

2.1F2F3step 1.1∎

The event ${∣ M_{n} - M_{0} ∣ \geq t}$ is the union of the upper and lower tail events. F2 gives twice the bound in step 1.1. If all $c_{k} = 0$ , all increments vanish almost surely and the event is empty. Zero-width individual terms otherwise simply contribute zero to the sum. AC is used exactly as stated in F3.

Depends on

Used by

Azuma bound for simple random walk Example

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Roch, Notes 20: Azuma's Inequality, Theorem 20.8, pp. 3–4 (standard reference, not scraped)