TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Convex functions of martingales are submartingales

Statement

Assume AC. If $M$ is a real martingale and $ϕ : R \to R$ is finite convex with $ϕ (M_{n}) \in L^{1} (P)$ for every $n$ , then $(ϕ (M_{n}))$ is a submartingale. If $X$ is instead a submartingale and $ϕ$ is also nondecreasing, the same conclusion holds provided $ϕ (X_{n}) \in L^{1} (P)$ for every $n$ .

Facts & Assumptions

Given: The hypotheses and conventions in the statement.

[F1]

A finite convex real function is Borel measurable. Convex functions have countable supporting line representations.

[F2]

Composition with a Borel outer function preserves measurability. Composition with a Borel measurable outer map preserves measurability.

[F3]

Conditional Jensen applies when the real input and its finite convex image are integrable. Conditional jensen inequality.

[F4]

AC supplies the inherited conditional-expectation existence and any stated choice of versions. The Axiom of Choice.

Proof

technique · direct

1.1

The function $ϕ$ is Borel, so each $ϕ (M_{n})$ is $F_{n}$ -measurable. Integrability of the image is an explicit assumption. At time $n + 1$ the input $M_{n + 1}$ is integrable by the martingale definition and its image is integrable by hypothesis. Conditional Jensen therefore gives $E [ϕ (M_{n + 1}) ∣ F_{n}] \geq ϕ (E [M_{n + 1} ∣ F_{n}]) = ϕ (M_{n})$ a.s. These are exactly the submartingale inequalities Martingale submartingale and supermartingale.

givenF1F2F3

2.1

For a submartingale $X$ , measurability and integrability of $ϕ (X_{n})$ hold by the same argument. Jensen and the nondecreasing hypothesis give $E [ϕ (X_{n + 1}) ∣ F_{n}] \geq ϕ (E [X_{n + 1} ∣ F_{n}]) \geq ϕ (X_{n})$ a.s. Monotonicity is applied to the submartingale inequality at that fixed time. AC is inherited from conditional Jensen and the CE existence used in both computations. No convex image of an arbitrary integrable variable is presumed integrable.

givenF1F2F3F4step 1.1∎

Depends on

Used by

Absolute value and powers of a martingale are submartingales Corollary

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

Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)