LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Conditional joint-typicality packing bound for finite alphabets

Statement

Let $(V, U, Z)$ have any joint law $P_{V U Z}$ on finite alphabets. Fix $η > 0$ . For every sufficiently small strong-typicality tolerance, every sufficiently large $n$ , and every fixed $(v^{n}, z^{n})$ typical for the marginal $P_{V Z}$ , an independent candidate word drawn from the product law $P_{U ∣ V}^{\otimes n} (\cdot ∣ v^{n})$ satisfies $\Pr {(v^{n}, U^{n}, z^{n}) is jointly typical for P_{V U Z}} \leq e^{- n (I_{P} (U; Z ∣ V) - η)} .$ The conclusion includes the case that $V$ is constant, giving the exponent $I_{P} (U; Z)$ .

Facts & Assumptions

Given: A finite joint law $P_{V U Z}$ , $η > 0$ , and fixed typical $(v^{n}, z^{n})$ .

[F1]

Finite strongly typical sets with support restriction: typicality restricts empirical distributions to a finite-alphabet neighborhood of the target law and excludes unsupported atoms.

[F2]

Finite conditional mutual information: $I (U; Z ∣ V)$ is the finite conditional-KL average in nats.

Proof

1.1F1F2givenalgebra

For any jointly typical supported candidate $u^{n}$ , $\ln \frac{P_{U ∣ V}^{\otimes n} (u^{n} ∣ v^{n})}{P_{U ∣ V, Z}^{\otimes n} (u^{n} ∣ v^{n}, z^{n})} = n \sum_{v, u, z} {\hat{P}}_{v^{n} u^{n} z^{n}} (v, u, z) \ln \frac{P (u ∣ v)}{P (u ∣ v, z)} .$ The logarithms are finite on the positive target support. At $P_{V U Z}$ the finite sum equals $- I_{P} (U; Z ∣ V)$ by [F2], and it is linear in the empirical probabilities. Choose the typicality tolerance small enough that every jointly typical triple has this sum at most $- I_{P} (U; Z ∣ V) + η$ .

2.1F1step 1.1algebra∎

Sum the likelihood-ratio expression of step 1.1 over the jointly typical candidates. The target conditional law is a probability vector, so its mass on that set is at most one; the exponential likelihood ratio is at most $e^{- n (I_{P} (U; Z ∣ V) - η)}$ . This gives the claimed bound. Joint typicality excludes unsupported triples; on each supported target triple both conditional probabilities in the ratio are positive.

Depends on

Used by

Decode-forward inner bound for a finite relay channel Theorem

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

Abbas El Gamal and Young-Han Kim, Lecture Notes on Network Information Theory, arXiv:1001.3404v4 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, Cambridge University Press prepublication book (standard reference, not scraped)