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A petite drift enlargement is small under aperiodicity

Statement

Let $P$ be countably generated, $ϕ$ -irreducible and aperiodic with invariant probability $π$ . Suppose $D$ is petite with probability $ν$ and multiplier $α > 0$ , and $C \subseteq D$ is $(m, ϵ, ν)$ -small for this same $ν$ , with $ν (C) > 0$ and $ν ≪ π$ . Then $D$ is small at some common time.

Facts & Assumptions

Given: The kernel, irreducibility, aperiodicity, invariant probability, petite inequality on $D$ , and same-measure small-set minorization on $C$ .

[F1]

Petite sets for a Markov kernel: $\sum_{n \geq 1} a_{n} P^{n} (x, \cdot) \geq α ν (\cdot)$ on $D$ .

[F2]

Small-set minorization and geometric drift: $P^{m} (y, \cdot) \geq ϵ ν$ on $C$ .

[F3]

Aperiodicity makes minorization times eventually consecutive: once every state can reach a minorization into $ν ≪ π$ , $ν P^{n} \geq c_{n} ν$ for every sufficiently large $n$ .

[F4]

Phi-irreducibility for a general-state Markov kernel: every positive- $ϕ$ set is accessible from every state.

[F5]

Discrete strong Markov property: the chain restarts from its state at the first entrance time.

[F6]

Invariant and stationary distribution for a Markov kernel: $π P^{n} = π$ for every $n \geq 0$ .

[A1]

The Axiom of Choice: the canonical path/stopping-time construction uses AC as specified by [F5].

Proof

Given: The hypotheses and facts [F1]–[F5], with [A1].

1.1F1given

The petite inequality on $C$ gives $\sum_{n} a_{n} P^{n} (x, C) \geq α ν (C) > 0$ for every $x \in D$ . Choose $M$ so $\sum_{n > M} a_{n} < α ν (C) / 2$ . Since $P^{n} (x, C) \leq \Pr_{x} (τ_{C} \leq n) \leq \Pr_{x} (τ_{C} \leq M)$ for $n \leq M$ , the truncated weighted sum gives $\Pr_{x} (τ_{C} \leq M) \geq q : = α ν (C) / 2$ uniformly on $D$ .

2.1F2F3F4F6step 1.1

Since $ν (C) > 0$ and $ν ≪ π$ , $π (C) > 0$ . To see that every state has positive probability of reaching $C$ , let $H = {x \in C^{c} : P^{n} (x, C) = 0 for every n \geq 1}$ . Kernel iteration makes $H$ measurable. If $x \in H$ and $P (x, H^{c}) > 0$ , the countable union $H^{c} = C \cup ⋃_{n \geq 1} {y : P^{n} (y, C) > 0}$ has a member of positive $P (x, \cdot)$ -mass; integrating the corresponding positive function gives $P^{n + 1} (x, C) > 0$ , a contradiction. Thus $P (x, H) = 1$ for every $x \in H$ . Suppose $H \neq \emptyset$ and fix $x \in H$ . Irreducibility then forces $ϕ (H^{c}) = 0$ , since no point of $H$ can reach $H^{c}$ ; hence $ϕ (H) > 0$ . For every $n \geq 1$ , invariance and $P^{n} (x, H) = 1$ on $H$ give $\int_{H^{c}} P^{n} (x, H) π (d x) = 0$ . Irreducibility says every $x \in H^{c}$ has $P^{n} (x, H) > 0$ for some $n$ , so the countable union of these $π$ -null level sets is $H^{c}$ and $π (H^{c}) = 0$ , contradicting $π (C) > 0$ . Therefore $H = \emptyset$ . For $x \notin C$ some positive time reaches $C$ by the definition of $H$ ; for $x \in C$ , [F2] gives $P^{m} (x, C) \geq ϵ ν (C) > 0$ . Thus for each state there is $n (x) \geq 1$ with $P^{n (x)} (x, C) > 0$ ; then $P^{n (x) + m} (x, \cdot) \geq ϵ P^{n (x)} (x, C) ν (\cdot)$ . By [F3], $ν P^{t} \geq c_{t} ν$ for every $t \geq N_{0}$ . Choose $L > M + m + N_{0}$ . For every $0 \leq s \leq M$ and $y \in C$ , $P^{L - s} (y, \cdot) \geq ϵ ν P^{L - s - m} (\cdot) \geq ϵ c_{L - s - m} ν (\cdot) .$ The minimum $c > 0$ over these finitely many $s$ is positive.

3.1F5A1step 1.1step 2.1∎

Decompose at the first hitting time $τ_{C} = s \leq M$ . The strong Markov property and step 2.1 give $P^{L} (x, \cdot) \geq c ν (\cdot) \Pr_{x} (τ_{C} \leq M) \geq c q ν (\cdot)$ for every $x \in D$ . This is one fixed-time minorization, so $D$ is small.

Depends on

Used by

Geometric drift yields a bounded small set and bivariate drift Lemma

Dependency tree · two levels

29 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)