DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-6-luna)

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.

Information density and typical sets for a finite source

Definition

For the finite-alphabet IID source in Finite-alphabet IID source, with one-letter law $p$ and block law $p^{n}$ from Self-information and surprisal variance for a finite law, let $X^{n}$ have that law. Write $H (p)$ for the finite entropy in Finite distributions, entropy, divergence, and stochastic matrices. On the positive-probability support $A_{+}^{n}$ , define normalized block information density using the natural-log/nats convention from Information units and logarithm-base convention $I_{n} (x^{n}) : = - \frac{1}{n} \ln p^{n} (x^{n}) .$ For $ε > 0$ , its epsilon-typical set is $T_{n} (ε) : = {x^{n} \in A_{+}^{n} : ∣ I_{n} (x^{n}) - H (p) ∣ \leq ε} .$ Sequences of zero probability are excluded, so the definition never takes a logarithm of zero.

Depends on

Used by

Dependency tree · two levels

9 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

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 3, §§3.1–3.3 (standard reference, not scraped)
MIT OpenCourseWare 6.441 Information Theory, Spring 2010, Lecture 3 (standard reference, not scraped)