DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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.

Adversarial sample replacement

Definition

For a statistical experiment ${P_{θ}}$ from Statistical experiments, sample size $N \geq 1$ , and replacement fraction $0 \leq η \leq 1$ , equip $X^{N}$ with its finite product sigma-algebra (The product sigma-algebra and its finite iterates). A sample-replacement coupling is a pair of $X^{N}$ -valued random elements on a common probability space (Random elements and real random variables) $(X, Y)$ such that $X_{1}, \dots, X_{N}$ are iid with law $P_{θ}$ under Identical distribution and IID families and

\sum_{i = 1}^{N} 1 {X_{i} \neq Y_{i}} \leq ⌊ η N ⌋

almost surely. Assume the diagonal $Δ_{X} = {(x, x) : x \in X}$ belongs to the product sigma-algebra $A \otimes A$ (The product sigma-algebra and its finite iterates), so each mismatch indicator is measurable. The joint law may depend on the entire clean sample; the observed vector $Y$ need not be iid.

Depends on

Used by

Half-sample replacement prevents covariance-only mean recovery Counterexample
Contamination-robust expected risk and tail risk Definition
Bounded sample-mean change under replacements and a common corrupted sample Example
Exact recovery by the multivariate trimmed mean at zero covariance Lemma
Correct-scale slab bound for multivariate trimming Proposition
Multivariate trimmed-mean bound under sample replacement Theorem
Univariate trimmed-mean bound under sample replacement Theorem

Dependency tree · two levels

14 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

P. B. Stark, Statistics 240 Lecture Notes, Part 9: Robustness and Related Topics, §1.2, PDF pp. 10–12 (standard reference, not scraped)