DefinitionDefinition: 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.

Finite-past information densities

Definition

Consider a stationary finite-alphabet source $(X_{i})_{i \in Z}$ with law $P$ and alphabet $A$ .

For $k \geq 0$ , let $F_{k} = σ (X_{- k}, \dots, X_{- 1})$ , with $F_{0}$ trivial. For each $a \in A$ , define $p_{k} (a) = P (X_{0} = a ∣ F_{k})$ by averaging over the finite atoms of $F_{k}$ , and put $g_{k} = - \log_{2} p_{k} (X_{0})$ on the event $p_{k} (X_{0}) > 0$ , with arbitrary value on its null complement.

The selected probability is positive almost surely: on each atom $B \in F_{k}$ with $P (B) > 0$ , $P (X_{0} = a ∣ B) = 0$ implies $P (B \cap {X_{0} = a}) = 0$ . Thus $g_{k}$ is finite almost surely; Finite-alphabet stationary source identifies the same distributions after shifting any finite window.

Facts & Assumptions

Given: A stationary finite-alphabet source $(X_{i})_{i \in Z}$ with law $P$ and alphabet $A$ .

[F1]

Finite-alphabet stationary source: the shift on $A^{Z}$ preserves $P$ . [F2] Finite-block information and entropy in bits: for $t > 0$ , $\log_{2} t = \ln t / \ln 2$ ; the information of a positive-probability block is its negative bit logarithm.

Proof

Given: The source and conventions [F1]–[F2].

1.1givenalgebra

2.1F1step 1.1algebra∎

The selected probability is positive almost surely: on each atom $B \in F_{k}$ with $P (B) > 0$ , $P (X_{0} = a ∣ B) = 0$ implies $P (B \cap {X_{0} = a}) = 0$ . Thus $g_{k}$ is finite almost surely; [F1] identifies the same distributions after shifting any finite window.

Depends on

Used by

Dependency tree · two levels

7 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

Robert M. Gray, Entropy and Information Theory, corrected edition (standard reference, not scraped)