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Measurable coordinatewise functions preserve independence
Statement
Let be an independent family of random elements . For each , let be measurable. Then the family is independent.
Facts & Assumptions
Given: Independent random elements and measurable maps as in the Statement.
Measurable outer maps preserve measurability under composition. (Composition with a Borel measurable outer map preserves measurability)
Independence of random elements is equivalent to the rectangle criterion. (Independent random elements are characterized by finite rectangle probabilities)
Proof
By [L1], each composite is again a random element.
Fix a finite list of distinct indices and measurable sets . Then for every , so [L2] applied to the independent family gives
Step 1.2 is exactly the rectangle criterion for the family , so [L2] shows that this family is independent.
Depends on
Used by
- Existence and scaling of d-dimensional Brownian motion Corollary
- Kolmogorov two-series sufficiency Corollary
- Matching minimax rate in a Gaussian orthogonal sequence model Corollary
- Weak law does not imply strong law Counterexample
- Antithetic randomized sampling Definition
- d-dimensional Brownian motion Definition
- Euler discretization bias and path-to-path variance Example
- Functions of disjoint independent coordinate blocks remain independent Example
- Radial second moment of multidimensional Brownian motion Example
- Sample variance from a multivariate delta method Example
- Strong law for empirical indicator averages Example
- Exact block budget for the covariance geometric-median pilot Lemma
- Finite-index bootstrap draws are conditionally IID from the empirical law Lemma
- Finite-second-moment IID sample covariance converges Lemma
- Independent centered Hilbert-valued elements have zero expected inner product Lemma
- Independent-copy symmetrization of random series Lemma
- IV observed contrasts as potential-outcome contrasts Lemma
- Jointly Gaussian blocks with zero cross-covariance are independent, including singular blocks Lemma
- KL prior mass gives a posterior denominator lower bound in probability Lemma
- KL-neighborhood prior mass lower-bounds the likelihood denominator Lemma
- Local KL and log-square mass bounds a non-IID posterior denominator Lemma
- Long-run covariance of an L2-m-approximable shift is trace class Lemma
- Marginal covariance of the linear mixed model Lemma
- Necessity of the truncated mean and variance conditions Lemma
- One-sided maximal inequality for symmetric independent sums Lemma
- Sieve metric entropy produces uniform exponential tests Lemma
- The standard Wishart inverse quadratic form has an exact Fisher law Lemma
- Uncorrelated jointly Gaussian blocks are independent Lemma
- Weighted overlapping-block moment and covariance limits Lemma
- A Gaussian contrast is independent of its orthogonal residual Theorem
- Antithetic variance and a monotone sufficient condition Theorem
- Asymptotic linearity and normality of iid Z-estimators Theorem
- Conditional Gaussian law of random effects Theorem
- Consistency of the IID Monte Carlo sample variance Theorem
- Dense sample covariance concentration for sub-Gaussian vectors Theorem
- Empirical measures of iid euclidean samples converge weakly Theorem
- Entrywise deviation for sample-centered covariance under directional Gaussian tails Theorem
- Etemadi strong law for pairwise independent iid variables Theorem
- Exact confidence and prediction intervals in Gaussian regression Theorem
- Exact t and partial F inference in full-rank Gaussian regression Theorem
…and 29 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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Theorem 2.1.10 (standard reference, not scraped)
- S. R. S. Varadhan, Probability Theory, Section 3.1 (standard reference, not scraped)