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

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Doob L1 maximal inequality

Statement

Assume AC. If $X$ is a nonnegative submartingale, $λ > 0$ , and $N \in N_{0}$ , then $λ P ⁣ (\max_{0 \leq k \leq N} X_{k} \geq λ) \leq E ⁣ [X_{N} 1_{{\max_{k \leq N} X_{k} \geq λ}}] \leq E X_{N} .$

Facts & Assumptions

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

[F1]

Martingale submartingale and supermartingale makes each first-crossing event measurable at its crossing time.

[F2]

Multistep martingale characterization gives $X_{k} \leq E [X_{N} ∣ F_{k}]$ for $k \leq N$ .

[F3]

Conditional expectation as an ae class supplies the integral identity on $F_{k}$ events, and The Lebesgue integral is linear on $L^{1} (μ)$ sums the finite partition.

[F4]

The Axiom of Choice is used only through the chosen conditional-expectation representatives in F2, F3.

Proof

1.1F1

Define the disjoint first-crossing events $A_{k} = {X_{0} < λ, \dots, X_{k - 1} < λ, X_{k} \geq λ}, 0 \leq k \leq N,$ with the preceding string empty for $k = 0$ . Each $A_{k} \in F_{k}$ , and their union is $A = {\max_{j \leq N} X_{j} \geq λ}$ .

1.2F2F3

On $A_{k}$ , $X_{k} \geq λ$ . By F2 and the conditional-expectation identity, $λ P (A_{k}) \leq E [X_{k} 1_{A_{k}}] \leq E [X_{N} 1_{A_{k}}] .$

2.1F3F4step 1.1step 1.2∎

Sum over the finite disjoint partition to get $λ P (A) \leq E [X_{N} 1_{A}] .$ Since $X_{N} \geq 0$ , the latter is at most $E X_{N}$ . This finite-time proof does not presuppose stopping-time or optional-sampling results. AC has exactly the inherited use in F4.

Depends on

Used by

An unbounded stopped exponential martingale needs uniform integrability Counterexample
Doob maximal bounds for a centered random walk Example
Brownian-filtration martingale representation Theorem
Doob Lp maximal inequality Theorem

Dependency tree · two levels

19 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

van der Vaart, Martingales, Diffusions and Financial Mathematics, Lemma 2.43, p. 22 (standard reference, not scraped)