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.
Identical distribution and IID families
Definition
Let be random elements with the same measurable target . They are identically distributed if for all and , that is, their laws in Law or distribution of a random element agree. They are independent and identically distributed (IID) if, in addition, the whole family is independent in Independent random elements. Independence means mutual independence, not merely pairwise independence. No moment assumption is part of either definition. The empty family satisfies these universal conditions vacuously.
Definition references
Depends on
Used by
- Aalen–Johansen cumulative-incidence estimator Algorithm
- Finite-simulation plug-in synthetic-likelihood evaluation Algorithm
- Independent-pilot cross-entropy proposal fitting and production Algorithm
- Ordinary importance sampling Algorithm
- Rejection sampling Algorithm
- Self-normalized importance sampling Algorithm
- Size-n bootstrap from the empirical law Algorithm
- CLT for independent computation replicates Corollary
- ERM excess risk from Rademacher uniform deviation Corollary
- Finite-sample exactness under equal distributions Corollary
- Iid finite variance strong law Corollary
- IID finite-variance weak law Corollary
- Isotropic minimum-norm risk limits under a positive spectral gap Corollary
- Wald first equation under integrable stopping Corollary
- An exact M-estimator can fail without population separation Counterexample
- An extra square-root sample-size factor destroys standard-error studentization Counterexample
- Identifiability and KL support do not replace testability Counterexample
- Infinite-dimensional Gaussian random-walk reverse flow is singular Counterexample
- Local-linear estimator can lack a finite-sample expectation Counterexample
- Pointwise resampling misses the long-run variance of an AR(1) mean Counterexample
- Pointwise strong laws need not give a uniform law Counterexample
- Positive kernel second moment does not imply nondegenerate first projections Counterexample
- The claimed univariate clean-quantile count exponent is too strong Counterexample
- The uniform endpoint MLE has rate n, not square-root n Counterexample
- Adversarial sample replacement Definition
- Antithetic randomized sampling Definition
- Control-variate estimators Definition
- Cross-validation splits and training-size risk targets Definition
- Density targets and the kernel density estimator Definition
- Empirical, centered, and Rademacher processes Definition
- Finite-alphabet IID source Definition
- Huber contamination model Definition
- Ideal and finite-simulation synthetic likelihood Definition
- IID empirical bootstrap sample and mean statistic Definition
- IID Monte Carlo sample mean, variance, and standard error Definition
- Latent-class contrastive representation learning Definition
- M-estimators Definition
- m-out-of-n bootstrap and subsampling Definition
- Method agreement and limits of agreement Definition
- Minimax squared operator risk Definition
…and 172 more results.
Dependency tree · two levels
6 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
- Section 2.2.1, p. 58, IID paragraph (standard reference, not scraped)