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.
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- 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.
Transition matrices and n-step probabilities
Definition
Let be countable with sigma-algebra , and let be a probability kernel on it. For , set
where is the identity kernel, so . By Iterated transition kernels, each is a probability kernel. Since measures on countable discrete spaces are their singleton-weighted sums (Every measure on a countable discrete space is its weighted sum of Dirac measures), every row satisfies
No conditional-expectation version or choice function is used in this matrix definition.
Depends on
Used by
- Expected exit time solves the Poisson equation Corollary
- Higher-dimensional simple symmetric walks are transient Corollary
- One-dimensional simple symmetric walk is recurrent Corollary
- Two-dimensional simple symmetric walk is recurrent Corollary
- A stationary chain need not be ergodic Counterexample
- Different classes can have different recurrence types Counterexample
- Positive recurrence without aperiodicity does not imply total-variation convergence Counterexample
- Accessibility, communication, and irreducibility Definition
- Aperiodic irreducible chain Definition
- Finite column-stochastic matrices and probability vectors Definition
- Finite row-stochastic kernel Definition
- Finite-chain hitting time and survival law Definition
- Green kernel of a transient chain Definition
- Invariant and stationary distribution for a Markov kernel Definition
- Nonnegative kernel action and finite drift Definition
- Period of a state Definition
- Recurrent and transient states Definition
- Reducibility of a finite Markov chain Definition
- Reversible measure and detailed balance Definition
- Simple symmetric walk on the integer lattice Definition
- A finite periodic chain need not converge in total variation Example
- A periodic chain has Cesaro but not ordinary convergence Example
- A proposal with disconnected support Example
- A shared missed region in a finite funnel-shaped target Example
- A three-state random-walk Metropolis kernel Example
- A two-state independence sampler Example
- Birth–death recurrence through scale products Example
- Communicating classes in a four-state chain Example
- Empirical state frequencies converge to stationary masses Example
- Gambler’s ruin from harmonicity Example
- Green kernel of a biased integer walk Example
- Laziness makes an irreducible chain aperiodic Example
- Negative drift gives a finite mean small-set hit Example
- Period two on a bipartite graph Example
- Rejection creates a self-transition Example
- Stationary law of a finite birth-and-death chain Example
- Stationary law of a two-state chain Example
- Thinning lowers observed lag correlation but discards information Example
- Uniform law for a finite doubly stochastic matrix Example
- Detailed balance implies invariance Lemma
…and 20 more results.
Dependency tree · two levels
10 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
- Durrett, Probability: Theory and Examples, fifth edition (standard reference, not scraped)
- Levin, Peres and Wilmer, Markov Chains and Mixing Times, second edition (standard reference, not scraped)