DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Moments, variance, and covariance on a probability space

Definition

Let $X$ be a real random variable on a probability space.

For $p > 0$ , the $p$ th absolute moment of $X$ exists when $∣ X ∣^{p}$ is integrable, and is then $E [∣ X ∣^{p}] .$
If $X$ is integrable, its mean is $E [X]$ .
If $X$ is square-integrable, its variance is $Var (X) : = E [(X - E [X])^{2}] .$

If $X$ and $Y$ are square-integrable real random variables on the same probability space, their covariance is $Cov (X, Y) : = E [(X - E [X]) (Y - E [Y])] .$

Depends on

Expectation of a nonnegative or integrable random variable

Used by

Finite-simulation plug-in synthetic-likelihood evaluation Algorithm
Best affine prediction from one random variable Corollary
Chebyshev's inequality for random variables Corollary
CLT for independent computation replicates Corollary
Conditional variance decomposition Corollary
Independence forces covariance to vanish Corollary
Independent RQMC replicates estimate randomization variance Corollary
Studentized RQMC replicate means are asymptotically normal as R→∞ Corollary
The second-moment lower bound for positive probability Corollary
A light-tailed proposal can give ordinary importance sampling infinite variance Counterexample
A mixed-model covariance does not imply MAR dropout Counterexample
A stationary sticky chain with finite variance and no root-n CLT Counterexample
An extra square-root sample-size factor destroys standard-error studentization Counterexample
Average cluster size does not determine the unequal-size design effect Counterexample
Feller negligibility cannot be removed from the converse Counterexample
Finite variance does not imply a sub-Gaussian tail Counterexample
Pointwise bootstrap need not preserve pair dependence Counterexample
Pointwise resampling misses the long-run variance of an AR(1) mean Counterexample
Positive kernel second moment does not imply nondegenerate first projections Counterexample
Unequal error variances can make OLS fail to be BLUE Counterexample
Uniform moving support invalidates the naive score bound Counterexample
Additive measurement error and categorical misclassification models Definition
Antithetic randomized sampling Definition
Batch and measurement-process variation Definition
Bias and mean squared error Definition
Binding function and indirect-inference target Definition
Biological and technical replication model Definition
Control-variate estimators Definition
Covariance matrix of a finite-dimensional random vector Definition
Critical, detection, and quantification levels Definition
Finite-dimensional linear mixed model with explicit longitudinal roles Definition
First-order and total Sobol indices Definition
Functional-specific effective sample size Definition
Generalized least-squares estimator for known covariance 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
Lagged lattice covariance and variogram estimators Definition
Linear processes, white noise, and ARMA models Definition
Long-run variance for a stationary Markov chain Definition

…and 195 more results.

Dependency tree · two levels

4 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

Jean-Francois Le Gall, Integration, Probabilities and Stochastic Processes, Section 8.2.1 (standard reference, not scraped)
Rick Durrett, Probability: Theory and Examples, 5th ed., Section 1.6 (standard reference, not scraped)