Alphabeta Math
CounterexampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07
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.

Convergence in distribution need not be convergence in probability

Statement refuted

Convergence in distribution need not imply convergence in probability.

Facts & Assumptions

Given: A random variable X with P(X=1)=P(X=−1)=1/2, and Xn=−X.

[L1]

Distributional convergence is convergence of the corresponding CDFs at continuity points (Convergence in distribution for real random variables).

[L2]

Probability convergence makes every positive error probability vanish (Convergence in probability).

Counterexample

technique · direct
1.1

The symmetric two-point law of −X equals that of X, so FXn=FX for every n. Therefore Xn⇒X by [L1].

L1
2.1

But ∣Xn−X∣=2 almost surely, so P(∣Xn−X∣>1)=1 for every n. By [L2], Xn does not converge to X in probability.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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