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.

Stationary finite-block information rate-distortion function

Definition

Let $(X_{t})_{t \in Z}$ be a stationary source on a finite nonempty alphabet $X$ , let $\hat{X}$ be finite nonempty, and let $d : X \times \hat{X} \to [0, d_{\max}]$ be a bounded finite-valued distortion. For $n \geq 1$ and $D \geq 0$ , define

R_{n} (D) : = \frac{1}{n \ln 2} \min_{W_{n} : E d_{n} (X_{0}^{n - 1}, {\hat{X}}_{0}^{n - 1}) \leq D} I (X_{0}^{n - 1}; {\hat{X}}_{0}^{n - 1}) bits/symbol,

with value $+ \infty$ when the constraint is infeasible. The minimum is attained whenever feasible by Attainment of finite-alphabet rate-distortion minima applied to the block alphabets. Put

D_{\min} : = E \min_{\hat{x} \in \hat{X}} d (X_{0}, \hat{x}), D_{c} : = \min_{\hat{x} \in \hat{X}} E d (X_{0}, \hat{x}), R_{b l o c k} (D) : = \inf_{n \geq 1} R_{n} (D) .

The stationary-ergodic process information rate-distortion quantity is

R_{e r g} (D) : = \inf \underset{n \to \infty}{lim sup} \frac{1}{n \ln 2} I (X_{0}^{n - 1}; {\hat{X}}_{0}^{n - 1}),

where the infimum ranges over jointly stationary ergodic pairs of finite-alphabet processes with the given source marginal and $E d (X_{0}, {\hat{X}}_{0}) \leq D$ ; an empty infimum is $+ \infty$ .

Depends on

Used by

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Sources

Robert M. Gray, Entropy and Information Theory, corrected first edition (standard reference, not scraped)
Robert M. Gray, Source Coding Theory, first edition (standard reference, not scraped)