TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Whole-space minorization gives uniform geometric ergodicity

Statement

Let $P$ have invariant probability $π$ . If for some $m \geq 1$ , $0 < ϵ \leq 1$ , and probability $ν$ , $P^{m} (x, \cdot) \geq ϵ ν (\cdot) for every x,$ then for every $n \geq 0$ , $\sup_{x} ∥ P^{n} (x, \cdot) - π ∥_{T V} \leq (1 - ϵ)^{⌊ n / m ⌋} .$ Thus this chain is uniformly geometrically ergodic: there are common $M < \infty$ and $r < 1$ with $\sup_{x} ∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M r^{n}$ . No converse is asserted. When $ϵ = 1$ , the floor bound means $1$ for $n < m$ and $0$ for $n \geq m$ .

Facts & Assumptions

Given: The probability kernel, invariant probability, and whole-space minorization in the Statement.

[F1]

Small-set minorization and geometric drift: the stated lower bound is a probability-kernel minorization.

[F2]

Invariant and stationary distribution for a Markov kernel: $π P^{n} = π$ .

[F3]

Total variation distance for probability laws: total variation is the supremum over measurable events (up to the stated normalization).

[F4]

Total variation duality, kernel contraction, and coupling bound: a probability kernel contracts total variation.

Proof

Given: The hypotheses and facts [F1]–[F4].

1.1F1F3F4

Write $P^{m} (x, \cdot) = ϵ ν + (1 - ϵ) R_{x}$ ; if $ϵ = 1$ , take any probability residual because its coefficient is zero. For probabilities $μ, μ^{'}$ , the common term cancels, so $∥ μ P^{m} - μ^{'} P^{m} ∥_{T V} \leq (1 - ϵ) ∥ μ - μ^{'} ∥_{T V}$ by the event-supremum definition and kernel contraction.

2.1F2F4step 1.1algebra∎

Iterating step 1.1 $q = ⌊ n / m ⌋$ times and using invariance yields $∥ δ_{x} P^{q m} - π ∥_{T V} \leq (1 - ϵ)^{q}$ . Write $n = q m + r$ with $0 \leq r < m$ ; the remaining $P^{r}$ contracts total variation and leaves $π$ invariant, so the same bound holds at $n$ . Taking the supremum over $x$ proves the result.

Depends on

Used by

Uniform ergodicity gives no uniform practical mixing time Example

Dependency tree · two levels

13 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

Gareth O. Roberts and Jeffrey S. Rosenthal, General State Space Markov Chains and MCMC Algorithms (standard reference, not scraped)
Randal Douc, Eric Moulines, Pierre Priouret, and Philippe Soulier, Markov Chains (standard reference, not scraped)