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.
Tight family of probability measures
Definition
A family of Borel probabilities on a metric space S is tight if, for every , there is a compact such that for every . One K must work for the whole family. Compactness is Open cover, subcover, compact metric space, and compact subset of a metric space. The empty family is tight, witnessed by the empty compact set.
Depends on
Used by
- A nontight sequence with no probability law subsequence limit Counterexample
- Boundedness of first moments alone does not give uniform integrability Counterexample
- Pointwise limit discontinuous at zero signals mass escape Counterexample
- Tightness from a uniform moment bound Example
- Every borel probability on a polish space is tight Theorem
- Prokhorov tightness theorem on polish spaces Theorem
- Tightness from characteristic function equicontinuity at zero Theorem
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
- van Gaans, Definition 5.1, p. 14 (standard reference, not scraped)