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.
Independent but non-identically distributed coordinates
Example
Assume countable choice and dependent choice. On take the product whose th coordinate has Bernoulli parameter . Then the coordinates are independent but not identically distributed.
Verification
Given: Countable choice, dependent choice, and the stated Bernoulli laws.
The countable-product theorem supplies the product probability under the stated choice principles. The coordinate-independence corollary applies to this sequence of Bernoulli laws and gives independence.
while . Hence their laws differ, so the independent family is not identically distributed.
Depends on
- Assuming countable and dependent choice, countable products of arbitrary probability spaces
- Coordinate random elements of a countable product are independent
- Law or distribution of a random element
- The Axiom of Countable Choice ($\mathrm{AC}_\omega$)
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
26 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
- Kajino, Probability Theory, Theorem 3.65 (standard reference, not scraped)