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.
A metric for convergence in probability
Definition
For almost-sure equivalence classes of real random variables on a probability space, set The expectation is that of Expectation of a nonnegative or integrable random variable. The following theorem proves this is independent of representatives and is a metric; until then the display is a proposed formula on classes. Durrett's exercise uses the comparable bounded transform rather than ; the next theorem proves the displayed variant directly.
Depends on
Used by
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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Exercise 3.2.8 (comparison metric) (standard reference, not scraped)