Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pending
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.

Ergodic Shannon–McMillan–Breiman theorem

Statement

For a stationary ergodic finite-alphabet source, the componentwise information-rate limit is the constant entropy rate.

Facts & Assumptions

Given: A stationary ergodic finite-alphabet source with entropy rate h.

[F1]

Componentwise Shannon–McMillan–Breiman theorem: normalized block self-information converges almost surely to an invariant random variable of mean h.

[F2]

Equivalent invariant-set and invariant-function criteria for ergodicity: every finite-valued invariant measurable function in an ergodic probability system is almost surely constant.

Proof

Given: The source and [F1]–[F2].

1.1F1F2given

By [F1], the normalized block information converges almost surely to an invariant G∗ with EG∗=h. Since the source is ergodic, [F2] makes G∗ a constant almost surely.

2.1F1step 1.1algebra∎

The expectation of a constant equals that constant, so G∗=h almost surely. Therefore −n−1log⁡2P(X0n−1)→h almost surely. For h=0 this says the normalized information tends to zero; no positive-rate assumption is used.

Depends on

Used by

Dependency tree · two levels

18 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