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Phi-irreducibility for a general-state Markov kernel
Definition
Let be a countably generated measurable space (Measurable spaces and measurable sets), a probability kernel (Measure kernel and probability kernel), and a nonzero -finite measure (Finite, sigma-finite, and semifinite measures) on . Write for the -fold kernel iterate. The kernel is -irreducible if, for every with and every , there is an integer such that . Equivalently, the sum of these nonnegative probabilities over is positive. No recurrence or invariant law is part of this definition.
Depends on
Used by
- A stationary sticky chain with finite variance and no root-n CLT Counterexample
- Aperiodicity for a general-state Markov kernel Definition
- Harris recurrence relative to an irreducibility measure Definition
- 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 countably generated phi-irreducible chain has a positive-measure small set Lemma
- 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
- Geometric Markov chains admit delayed adjacent-iid split tours with spare moments Lemma
- Harris geometric ergodicity admits an integrable total-variation envelope Lemma
- Invariant mass and almost-sure hitting of accessible sets Lemma
- Iterated kernel densities retain convolution lower bounds 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
11 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)