DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

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Polynomial ergodicity with an integrable rate envelope

Definition

Let $P$ be a probability kernel with invariant probability $π$ . For $m > 0$ , say that $P$ has polynomial total-variation rate of order $m$ with an integrable envelope if there is a measurable $M : X \to [0, \infty)$ with $π (M) < \infty$ such that $∥ P^{n} (x, \cdot) - π ∥_{T V} \leq M (x) n^{- m} (x \in X, n \geq 1) .$

This is the specific envelope condition used with the polynomial-rate CLT cases on this page. It does not claim that every definition of polynomial ergodicity in the literature includes integrability of $M$ .

Depends on

Used by

Sufficient MCMC CLT regimes Theorem

Dependency tree · two levels

17 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

Galin L. Jones, On the Markov Chain Central Limit Theorem, §2 and Theorem 9 (standard reference, not scraped)