Alphabeta MathHow statement and proof provenance work
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Aperiodicity for a general-state Markov kernel
Definition
Let be a -irreducible probability kernel (Phi-irreducibility for a general-state Markov kernel) with invariant probability (Invariant and stationary distribution for a Markov kernel). The chain is aperiodic if there do not exist an integer and pairwise disjoint measurable sets such that and Invariance then gives for every . The sets need not cover the full state space; the definition rules out any positive-mass cyclic part. Aperiodicity does not itself assert convergence.
Depends on
Used by
- A finite periodic chain need not converge in total variation Example
- A phi-irreducible chain can fail to converge from a pi-null start Example
- A sticky general-state chain can converge without a geometric rate Example
- Drift and minorization certify geometric ergodicity Example
- Uniform ergodicity gives no uniform practical mixing time Example
- A petite drift enlargement is small under aperiodicity Lemma
- Aperiodicity makes minorization times eventually consecutive Lemma
- Geometric drift yields a bounded small set and bivariate drift Lemma
- Geometric ergodicity and reversibility give exponential rho-mixing Lemma
- Harris geometric ergodicity admits an integrable total-variation envelope Lemma
- Small-set coupling yields almost-everywhere total-variation convergence Lemma
- General-state Markov-chain convergence under recurrence conditions Theorem
- Geometric ergodicity under drift and small-set minorization Theorem
- Sufficient MCMC CLT regimes Theorem
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)