LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-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.

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The finite uniformized matrix is column stochastic

Statement

Let $Q$ be a finite column generator and let $ν \geq \max_{j < n} r_{j}$ . If $ν > 0$ , then

$P : = I + Q / ν$

has nonnegative entries and every column sums to one. If $ν = 0$ , then $Q = 0$ .

Facts & Assumptions

Given: A finite column generator $Q = (q_{i j})_{i, j < n}$ , its exit rates $r_{j}$ , and $ν \geq \max_{j} r_{j}$ .

[F1]

Finite column generator: $q_{i j} \geq 0$ for $i \neq j$ , $r_{j} = - q_{j j} = \sum_{i \neq j} q_{i j}$ , and $\sum_{i} q_{i j} = 0$ for every column $j$ .

Proof

technique · direct

1.1givenF1algebra

Suppose $ν > 0$ . For $i \neq j$ , $P_{i j} = q_{i j} / ν \geq 0$ by [F1]. For a diagonal entry, $P_{j j} = 1 - r_{j} / ν \geq 0$ because $r_{j} \leq ν$ . Thus every entry is nonnegative.

2.1F1step 1.1algebra

For each source column $j$ , [F1] gives $\sum_{i} P_{i j} = \sum_{i} (δ_{i j} + q_{i j} / ν) = 1 + ν^{- 1} \sum_{i} q_{i j} = 1$ . Hence $P$ is column stochastic.

3.1givenF1algebra∎

If $ν = 0$ , the bound $0 \leq r_{j} \leq ν$ gives $r_{j} = 0$ for every $j$ . Each finite sum $r_{j} = \sum_{i \neq j} q_{i j}$ consists of nonnegative terms, so each off-diagonal entry in column $j$ is zero; the column-sum condition then gives $q_{j j} = 0$ . Therefore $Q = 0$ .

Depends on

Finite column generator

Used by

Dependency tree · two levels

3 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

Ward Whitt, Continuous-Time Markov Chains, full lecture notes (2006) (standard reference, not scraped)
Brian Munsky and Mustafa Khammash, The Finite State Projection Algorithm for the Solution of the Chemical Master Equation (standard reference, not scraped)