LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generated

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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.

Random and systematic scan kernels are well-defined

Statement

Let $m \geq 1$ , let $(S, S)$ be one finite-product state space with target probability $π$ , and let $P_{1}, \dots, P_{m} : S \to S$ be exact Gibbs block-update probability kernels for that same $π$ , as in Random and systematic Gibbs scan schemes. If weights $w_{j} \geq 0$ with $\sum_{j = 1}^{m} w_{j} = 1$ are supplied, the random-scan branch defines a probability kernel on $S$ . Independently of any weights, the systematic-scan branch defines a probability kernel on $S$ .

Facts & Assumptions

Given: The common finite-product state space, target probability, and finite list of exact block-update probability kernels in the Statement; weights are additionally supplied only for the random-scan branch.

[F1]

Each block update is a probability kernel. The Gibbs block update is a probability kernel.

[F2]

The random-scan evaluation is the finite weighted sum $\sum_{j} w_{j} P_{j} (x, A)$ . Random and systematic Gibbs scan schemes.

[F3]

Composition in chronological order is $(K L) (x, A) = \int L (y, A) K (x, d y)$ and finite composition of probability kernels is well-defined. Composition of probability kernels, Kernel composition is well defined and associative.

Proof

Given: The hypotheses in the Statement and facts [F1]–[F3].

1.1F1F2algebra

For each measurable $A$ , the finite weighted sum in [F2] is measurable in $x$ by [F1], and it is nonnegative. For pairwise disjoint $A_{n}$ , countable additivity of each $P_{j} (x, \cdot)$ gives $\sum_{j} w_{j} P_{j} (x, ⋃_{n} A_{n}) = \sum_{j} w_{j} \sum_{n} P_{j} (x, A_{n}) = \sum_{n} \sum_{j} w_{j} P_{j} (x, A_{n})$ ; the interchange is valid because the sum over $j$ is finite and every term is nonnegative. Its total mass is $\sum_{j} w_{j} P_{j} (x, S) = \sum_{j} w_{j} = 1$ . Hence the random-scan kernel is a probability kernel.

2.1F1F3construct∎

Each $P_{j}$ is a probability kernel by [F1]. Repeated application of the composition result in [F3] shows that $P_{1} \dots P_{m}$ is a probability kernel; associativity makes this finite product independent of parenthesization. This is the full-sweep transition kernel.

Depends on

Used by

A systematic Gibbs sweep preserves the target Theorem
Random-scan Gibbs is reversible and invariant Theorem

Cited to discharge well-definedness by Random and systematic Gibbs scan schemes.

Dependency tree · two levels

14 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

Art B. Owen, Monte Carlo Theory, Methods and Examples, Chapter 12 (standard reference, not scraped)
Gareth O. Roberts and Jeffrey S. Rosenthal, General State Space Markov Chains and MCMC Algorithms (standard reference, not scraped)