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.
Convergence in probability
Definition
For real random variables and on one probability space, write in probability when, for every , This is precisely Convergence in measure for the probability measure.
Depends on
Used by
- Convergence in distribution need not be convergence in probability Counterexample
- Convergence in probability need not be almost sure Counterexample
- Convergence in probability need not imply Lᵖ convergence Counterexample
- A probability-convergent sequence with a prescribed fast almost-sure subsequence Example
- Pairing preserves convergence in probability Lemma
- Almost-sure convergence implies convergence in probability Theorem
- An almost-surely convergent subsequence from convergence in probability Theorem
- Continuous maps preserve convergence in probability Theorem
- Convergence in distribution to a constant is convergence in probability Theorem
- Convergence in probability implies convergence in distribution Theorem
- Convergence in probability is metrized by d₀ Theorem
- Limits in probability are unique almost surely Theorem
- Lᵖ convergence implies convergence in probability Theorem
- Slutsky's theorem for real random variables Theorem
- Subsequence characterization of convergence in probability Theorem
- Uniform integrability plus convergence in probability implies L¹ convergence 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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Section 2.2 (standard reference, not scraped)