CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge 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.

Typical-set probability and cardinality bounds

Statement

For a finite-alphabet IID source and $ε > 0$ , let $T_{n} (ε)$ be its typical set and $δ_{n} : = P (X^{n} \notin T_{n} (ε))$ . Then $0 \leq δ_{n} \leq \frac{V (p)}{n ε^{2}}, ∣ T_{n} (ε) ∣ \leq e^{n (H (p) + ε)}, ∣ T_{n} (ε) ∣ \geq (1 - δ_{n}) e^{n (H (p) - ε)} .$ In particular $δ_{n} \to 0$ .

Facts & Assumptions

Given: A finite-alphabet IID source, its entropy $H (p)$ and surprisal variance $V (p)$ , and $ε > 0$ .

[F1]

Information density and typical sets for a finite source: $T_{n} (ε)$ consists of positive-probability sequences satisfying $∣ I_{n} (x^{n}) - H (p) ∣ \leq ε$ .

[F2]

Finite-alphabet IID information AEP with a variance bound: $P (∣ I_{n} (X^{n}) - H (p) ∣ \geq ε) \leq V (p) / (n ε^{2})$ .

Proof

Given: The source and $ε$ in the Statement.

1.1givenF1algebra

If $x^{n} \in T_{n} (ε)$ , [F1] gives $p^{n} (x^{n}) = e^{- n I_{n} (x^{n})} \geq e^{- n (H (p) + ε)}$ . Summing these probabilities over $T_{n} (ε)$ and using total mass one yields $∣ T_{n} (ε) ∣ e^{- n (H (p) + ε)} \leq 1$ , proving the upper bound.

1.2givenF1algebra

The other side of [F1] gives $p^{n} (x^{n}) \leq e^{- n (H (p) - ε)}$ for each typical sequence. Hence $P (X^{n} \in T_{n} (ε)) \leq ∣ T_{n} (ε) ∣ e^{- n (H (p) - ε)}$ ; rearranging and using $P (X^{n} \in T_{n} (ε)) = 1 - δ_{n}$ proves the lower bound.

2.1F2algebra∎

The complement of the typical set is contained in ${∣ I_{n} (X^{n}) - H (p) ∣ \geq ε}$ , so [F2] bounds $δ_{n}$ by $V (p) / (n ε^{2})$ . This tends to zero, including when $V (p) = 0$ .

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 (standard reference, not scraped)
MIT OpenCourseWare 6.441 Information Theory, Spring 2010, Lecture 3: AEP and typical sets (standard reference, not scraped)