LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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An invariant law integrates a drift function bounded on its small set

Statement

Let $π$ be invariant for $P$ , and suppose $P V \leq λ V + b 1_{C}$ with $V \geq 1$ , $0 < λ < 1$ , and $\sup_{C} V < \infty$ . Then $π V < \infty$ .

Facts & Assumptions

Given: The invariant probability and drift inequality in the Statement.

The supremum is interpreted in the extended reals, with $\sup \emptyset = - \infty$ ; the empty-set case is handled separately below.

[F1]

Small-set minorization and geometric drift: $V$ is finite-valued and $P V \leq λ V + b 1_{C}$ pointwise.

[F2]

Invariant and stationary distribution for a Markov kernel: $π P = π$ , including equality of integrals for bounded measurable functions.

[F3]

Measure kernel and probability kernel: $P (V \land N)$ is measurable and bounded by $N$ .

[F4]

Monotone convergence for the integral: $\int (V \land N) d π ↑ \int V d π$ .

Proof

Given: The hypotheses and facts [F1]–[F4].

1.1F1F2given

If $C = \emptyset$ , then $P V \leq λ V$ everywhere. For each integer $N \geq 1$ , $V_{N} = V \land N$ satisfies $P V_{N} - V_{N} \leq - (1 - λ) V$ on ${V \leq N}$ and $P V_{N} - V_{N} \leq 0$ on ${V > N}$ . Invariance for the bounded function $V_{N}$ gives integral zero, so $π ({V \leq N}) = 0$ . Since $V$ is finite-valued, these sets increase to the whole space, contradicting that $π$ is a probability. Thus no invariant probability exists in the empty-set case, and the implication is vacuous. Assume $C \neq \emptyset$ below.

1.2F1F3given

Set $V_{N} = V \land N$ . Since $V_{N} \leq V$ and $V_{N} \leq N$ , $P V_{N} \leq \min (P V, N)$ . On $C$ , $P V_{N} - V_{N} \leq λ \sup_{C} V + b$ . On $C^{c} \cap {V \leq N}$ it is at most $- (1 - λ) V$ , and on $C^{c} \cap {V > N}$ it is at most zero.

2.1F2step 1.2

Invariance for bounded $V_{N}$ gives $\int (P V_{N} - V_{N}) d π = 0$ . Using the three pointwise bounds from step 1.2, $(1 - λ) \int_{C^{c} \cap {V \leq N}} V d π \leq (λ \sup_{C} V + b) π (C) \leq λ \sup_{C} V + b .$

3.1F4step 2.1∎

The domains $C^{c} \cap {V \leq N}$ increase to $C^{c}$ ; monotone convergence bounds $\int_{C^{c}} V d π$ by the right side divided by $1 - λ$ . The integral over $C$ is at most $\sup_{C} V$ . Hence $π V < \infty$ .

Depends on

Used by

Geometric ergodicity under drift and small-set minorization Theorem

Dependency tree · two levels

26 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)
Randal Douc, Eric Moulines, Pierre Priouret, and Philippe Soulier, Markov Chains (standard reference, not scraped)