Alphabeta MathHow 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.
Canonical Markov chain on path space
Statement
Assume Choice. For every probability measure and probability kernel on an arbitrary measurable space , the canonical path space carries a unique probability under which the coordinate maps form a Markov chain with initial law and transition kernel .
Facts & Assumptions
Given: Choice, , , and as in the statement.
Ionescu--Tulcea constructs a unique countable-product law for specified history-dependent kernels, and its homogeneous last-coordinate specialization is a Markov chain. (Ionescu-Tulcea construction of a Markov chain)
Proof
In [F1], take for all , initial law , and [F1] The last display is a probability kernel because it is the composition of the measurable last-coordinate projection with each measurable evaluation of .
The theorem [F1] therefore gives a unique law on the product sigma-algebra and says that [F1, step 1.1] the coordinates are the required -chain. Choice is exactly the assumption of [F1]. The construction includes Dirac initial laws, one-point spaces, and ; an empty admits no probability and so cannot meet the hypotheses.
Depends on
Used by
- Higher-dimensional simple symmetric walks are transient Corollary
- One-dimensional simple symmetric walk is recurrent Corollary
- The post-hitting chain restarts from the hit state Corollary
- Two-dimensional simple symmetric walk is recurrent Corollary
- A sticky general-state chain can converge without a geometric rate Example
- Birth–death recurrence through scale products Example
- Gambler’s ruin from harmonicity Example
- Green kernel of a biased integer walk Example
- Random-mapping representation for a finite transition matrix Example
- A bivariate drift and minorization bound geometric coupling error Lemma
- A drift sublevel enlargement of a small set is petite Lemma
- A petite drift enlargement is small under aperiodicity Lemma
- Geometric Markov chains admit delayed adjacent-iid split tours with spare moments Lemma
- Geometric small-set renewal convergence gives an integrable exponential hitting moment Lemma
- Harris geometric ergodicity admits an integrable total-variation envelope Lemma
- Invariant mass and almost-sure hitting of accessible sets Lemma
- Small-set coupling yields almost-everywhere total-variation convergence Lemma
- Zero-variance geometric Markov sums have a uniform strong rate Lemma
- Convergence to stationarity for irreducible aperiodic positive-recurrent chains Theorem
- General-state Markov-chain convergence under recurrence conditions Theorem
- Lyapunov drift bound for hitting times Theorem
- Markov property for bounded future path functionals Theorem
- Poisson-equation martingale decomposition and MCMC CLT Theorem
- Recurrence and transience are class properties Theorem
- Strong consistency of overlapping batch means Theorem
- Time reversal of a stationary Markov chain Theorem
Dependency tree · two levels
14 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
- Shalizi, Building Infinite Processes from Finite-Dimensional Distributions, Theorem 33 (standard reference, not scraped)