LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-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.

Finite probability bounds for an empirical type

Statement

Let $A$ be a finite alphabet of size $k \geq 1$ , let $p$ be a probability vector on $A$ , and let $q \in P_{n} (A)$ . Then $(n + 1)^{- (k - 1)} e^{- n D (q ∥ p)} \leq p^{\otimes n} (T_{q}) \leq e^{- n D (q ∥ p)},$ with $e^{- n \cdot \infty} : = 0$ when $q$ is not supported on $p$ .

Facts & Assumptions

Given: A finite alphabet $A$ , a probability vector $p$ , a positive integer $n$ , and an attainable type $q$ .

[F1]

Exact probability of an empirical type class: $p^{\otimes n} (T_{q}) = ∣ T_{q} ∣ e^{- n [H (q) + D (q ∥ p)]}$ , with value zero if $D (q ∥ p) = + \infty$ .

[F2]

Polynomially sharp bounds for type-class cardinality: $(n + 1)^{- (k - 1)} e^{n H (q)} \leq ∣ T_{q} ∣ \leq e^{n H (q)}$ .

[F3]

Finite distributions, entropy, divergence, and stochastic matrices: finite relative entropy is $+ \infty$ exactly when positive mass of $q$ falls on a zero coordinate of $p$ .

Proof

1.1F1F2algebra

If $D (q ∥ p) < \infty$ , substitute [F1] and [F2] and cancel the finite factors $e^{n H (q)} e^{- n H (q)}$ to obtain both inequalities. The case $k = 1$ is included: then $q = p = (1)$ and all three quantities equal $1$ .

2.1F1F3step 1.1algebra∎

If $D (q ∥ p) = + \infty$ , [F1] and [F3] give $p^{\otimes n} (T_{q}) = 0$ , and both stated bounds are zero by convention.

Depends on

Used by

A finite Bernoulli large-deviation tail Example
Finite-alphabet Sanov bounds on the support simplex Theorem

Dependency tree · two levels

8 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, second edition, Chapter 11 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, Information Theory: From Coding to Learning, author-hosted book draft (standard reference, not scraped)