Alphabeta Math
CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10
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.

Iid strong law implies the weak law

Statement

For IID integrable real variables, Sn/n→EX1 in probability.

Facts & Assumptions

[F1]

Kolmogorov iid l1 strong law: For IID real (Xn)n≥1 with E∣X1∣<∞, Sn/n→μ=EX1 almost surely.

[F2]

Almost-sure convergence implies convergence in probability: If Xn→X almost surely, then Xn→X in probability.

Proof

Given: The objects, hypotheses and definitions in the statement. Its conclusions are to be established below.

1.1

F1 applies to the given IID integrable sequence and gives Sn/n→EX1 almost surely.

F1
2.1

The sample means and the constant limit are real random variables on that same probability space. Thus F2 applies to step 1.1 and gives the claimed probability convergence.

F2step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources