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.
Random elements and real random variables
Definition
Let be a probability space and let be a measurable space. A random element of is a measurable map in the sense of A measurable function between measurable spaces.
When and from The Borel sigma-algebra of a topological space, the map is a real random variable.
Depends on
Used by
- Dominating thinning for bounded time-varying propensities Algorithm
- Gillespie direct method with horizon and budget statuses Algorithm
- Inverse probability weighting for a causal mean Algorithm
- Minimum-intensity coupling of reaction networks Algorithm
- Next-reaction method by internal Poisson clocks Algorithm
- Standardization for a causal mean Algorithm
- Posterior RCD exists for a standard-Borel parameter Corollary
- A noncylinder path functional may fail to be measurable Counterexample
- Almost-sure finiteness does not imply integrability Counterexample
- Finite variance does not imply a sub-Gaussian tail Counterexample
- Uniform payoff convergence does not imply gradient convergence Counterexample
- A metric for convergence in probability Definition
- A simple Markovian mediation model Definition
- Adapted and integrable stochastic process Definition
- Additive measurement error and categorical misclassification models Definition
- Adversarial sample replacement Definition
- Almost-sure convergence of real random variables Definition
- Batch and measurement-process variation Definition
- Bayesian posterior as a regular conditional law Definition
- Binary instrumental-variable potential-outcome setup Definition
- Biological and technical replication model Definition
- Controlled and natural mediation effects Definition
- Convergence in probability Definition
- Covariance matrix of a finite-dimensional random vector Definition
- Critical, detection, and quantification levels Definition
- Discrete stopping time Definition
- Empirical, centered, and Rademacher processes Definition
- Exchangeable random elements Definition
- Expectation of a nonnegative or integrable random variable Definition
- Filtration-adapted stochastic-gradient oracles and Robbins–Monro updates Definition
- Finite conditional entropy Definition
- Finite conditional mutual information Definition
- Finite mutual information Definition
- Finite randomized treatment-assignment models Definition
- Finite-dimensional linear mixed model with explicit longitudinal roles Definition
- Fixed-design linear regression model Definition
- Full and observed data under a missingness pattern Definition
- Given countable nonexplosive pure-jump chain with finite rates Definition
- Independent random elements Definition
- L2-m-approximable causal Bernoulli shifts Definition
…and 77 more results.
Dependency tree · two levels
9 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
- S. R. S. Varadhan, Probability Theory, Definition 1.10 (standard reference, not scraped)
- J. R. Norris, Probability and Measure, Section 2.1 (standard reference, not scraped)