DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Initial distribution of a Markov chain

Definition

Under the Choice convention in the definition of a Markov chain, the initial distribution of $X$ is the probability measure $μ = L (X_{0}), μ (A) = P (X_{0} \in A), A \in E .$ The notation $P_{μ}$ denotes a specified law of a chain whose initial distribution is $μ$ ; it does not by itself choose a sample-space realization. When the initial state is fixed at $x$ , write $P_{x}$ for $P_{δ_{x}}$ and $E_{x}$ 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

Finite-dimensional laws of a Markov chain 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)