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 distribution of random elements
Definition
Random elements with values in the same metric space converge in distribution, written , if their laws from Law or distribution of a random element satisfy in Weak convergence of borel probability measures. They may be defined on different probability spaces. The definition specifies only their marginal laws.
Depends on
Used by
- A terminal weak limit does not imply a path-space weak limit Counterexample
- Asymptotic distribution Definition
- Donsker classes and asymptotic equicontinuity Definition
- Joint convergence in distribution and the normalized cycle-character observables Definition
- Local asymptotic normality for local experiments Definition
- Regular estimator under local alternatives Definition
- A degenerate weak limit with persistent expectation Example
- Compact-uniform LAN approximates compact-prior posteriors Lemma
- Central-limit convergence is only in distribution Remark
- A mean-one likelihood-ratio limit implies contiguity Theorem
- Asymptotic linearity and normality of iid Z-estimators Theorem
- Finite-dimensional stochastic-order calculus and tightness Theorem
- First- and second-order delta method for random vectors Theorem
- Fixed-order autoregressive OLS martingale limit Theorem
- Hilbert-valued CLT for L2-m-approximable Bernoulli shifts Theorem
- Le Cam's third lemma for a Gaussian likelihood-ratio limit Theorem
- Likelihood-ratio limit for local cones in a QMD model Theorem
- Misspecified M-estimator sandwich limit Theorem
- Multivariate iid central limit theorem Theorem
- Vector Slutsky theorem 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
- van Gaans, Definition 3.1 applied to laws (standard reference, not scraped)