ExampleConstruction: AI-generatedVerification: 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.

A perfectly dependent Bernoulli source has zero block information rate

Example

Fix $p \in (0, 1)$ and one Bernoulli $(p)$ random variable $Z$ . On its two-point probability space set $X_{i} = Z$ for every $i \geq 1$ . The variables have identical Bernoulli $(p)$ marginals but are dependent. Their normalized block self-information tends to $0$ on every outcome, whereas the marginal entropy $H (p) = - p \ln p - (1 - p) \ln (1 - p)$ is positive. Thus identical marginals alone do not imply the IID AEP conclusion.

Facts & Assumptions

Given: $p \in (0, 1)$ and $X_{i} = Z$ for one Bernoulli $(p)$ variable $Z$ .

[F1]

Self-information and surprisal variance for a finite law: For a finite law $q$ , self-information on its support is $- \ln q (x)$ .

[F2]

Finite distributions, entropy, divergence, and stochastic matrices: $H (p) = - \sum_{a : p (a) > 0} p (a) \ln p (a)$ with zero terms set to zero, in natural units.

[F3]

Finite-alphabet IID information AEP with a variance bound: For an IID finite-alphabet source, normalized block self-information converges in probability to its one-letter entropy.

Verification

Given: The example and notation above.

1.1givenF2algebra

Each $X_{i}$ has law $p$ on ${0, 1}$ , so $H (p) = - p \ln p - (1 - p) \ln (1 - p) > 0$ . For $0 < p < 1$ , both summands are positive. Also $P (X_{1} = 0, X_{2} = 0) = 1 - p \neq (1 - p)^{2}$ , so the variables are not independent and hence are not IID.

2.1F1step 1.1algebra

The length- $n$ block has only two possible values: $0^{n}$ with probability $1 - p$ and $1^{n}$ with probability $p$ . Therefore its normalized self-information is either $- \ln (1 - p) / n$ or $- \ln p / n$ , and is bounded by $\max {- \ln p, - \ln (1 - p)} / n$ . This tends to zero for both outcomes.

3.1F3step 1.1step 2.1∎

The limiting value $0$ differs from the positive marginal entropy in step 1.1. The IID conclusion [F3] therefore cannot be extended to all sequences with identical marginals after independence is removed.

Depends on

Used by

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Dependency tree · two levels

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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: Convergence and Asymptotic Equipartition Property (standard reference, not scraped)