CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The post-hitting chain restarts from the hit state

Statement

Assume Choice. For a measurable set $D \in E$ , define its hitting time $τ_{D} : = \inf {n \geq 0 : X_{n} \in D}, \inf \emptyset : = \infty .$ For every bounded measurable path functional $H$ , set both sides below to zero on ${τ_{D} = \infty}$ . Then $E [H (X_{τ_{D}}, X_{τ_{D} + 1}, \dots) ∣ F_{τ_{D}}] = E_{X_{τ_{D}}} H on {τ_{D} < \infty},$ in the explicit eventwise sense of the discrete strong Markov theorem. Thus, conditional on the information at the hit, the shifted chain has the canonical path law started from the hit state.

Facts & Assumptions

Given: Choice, a $K$ -chain and a measurable target $D$ .

[F1]

The discrete strong Markov theorem gives the eventwise future-functional identity at every stopping time, with both sides zero at infinity. (Discrete strong Markov property)

[F2]

For each $x$ , the canonical law $P_{x}$ is the unique path-space law of the $K$ -chain started from $x$ . (Canonical Markov chain on path space)

Proof

1.1

Adaptedness gives [given] ${τ_{D} \leq n} = ⋃_{k = 0}^{n} {X_{k} \in D} \in F_{n},$ so $τ_{D}$ is a stopping time. If $D = \emptyset$ , it is identically infinity; if $D = E$ , it is identically zero.

given

2.1

Apply [F1] to $τ_{D}$ . Its function [F1, F2, step 1.1] $h (x) = E_{x} H$ is exactly expectation under the canonical restarted law in [F2]. Therefore the conditional expectation of the shifted future equals $h (X_{τ_{D}})$ on the finite-hit event, with the slice-sum zero convention on its complement. Since this holds for every bounded measurable $H$ , it identifies the conditional path law, not only its one-time marginals. Constants zero and one check respectively zero mass and the finite-hit indicator. Choice is used by [F1]--[F2].

F1F2step 1.1∎

Depends on

Used by

Absorbing gambler's-ruin chain Example

Dependency tree · two levels

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Sources

Durrett, Probability: Theory and Examples, Section 5.2 (standard reference, not scraped)