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.
Initial distribution of a Markov chain
Definition
Under the Choice convention in the definition of a Markov chain, the initial distribution of is the probability measure The notation denotes a specified law of a chain whose initial distribution is ; it does not by itself choose a sample-space realization. When the initial state is fixed at , write for and for its expectation.
The definition includes Dirac, one-point, and arbitrary probability initial laws. There is no initial law on an empty state space, since no probability measure of total mass one exists there.
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
- Birth–death recurrence through scale products Example
- Gambler’s ruin from harmonicity Example
- Green kernel of a biased integer walk Example
- Geometric tail for hitting in a finite irreducible chain Lemma
- Bounded Dirichlet problem for hitting probabilities Theorem
- Equivalent criteria for recurrence and transience Theorem
- Finite-dimensional laws of a Markov chain Theorem
- First-step equations for nonnegative exit costs Theorem
- Hitting probability as minimal harmonic extension Theorem
- Invariant initial law makes a Markov chain stationary Theorem
- Lyapunov drift bound for hitting times Theorem
- Recurrence and transience are class properties Theorem
- Renewal decomposition at successive returns Theorem
- Superharmonic majorants bound exit costs Theorem
Dependency tree · two levels
7 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, Section 5.1 (standard reference, not scraped)