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Disjoint groups of an independent sigma-algebra family remain independent
Statement
Let be an independent family of sigma-algebras on a probability space, and let be pairwise disjoint index sets. For each , define
Then the sigma-algebras are independent.
Facts & Assumptions
Given: An independent family and pairwise disjoint index sets .
Independence of sigma-algebras means finite intersections of events from distinct member sigma-algebras satisfy the product formula. (Independent sigma-algebras and independent events)
Independent pi-systems containing the whole space generate independent sigma-algebras. (Independent pi-systems generate independent sigma-algebras)
Proof
For each , let be the class consisting of together with all finite intersections , where is finite and for every . Each contains and is closed under finite intersections, so it is a pi-system. Moreover by definition of .
Fix for each . Because the index sets are pairwise disjoint, the event is a finite intersection of events taken from distinct members of the original independent family. Hence [L1] gives
Step 1.2 says that the pi-systems are independent. Applying [L2] and using step 1.1 yields independence of for every .
Depends on
Used by
- Wald first equation under integrable stopping Corollary
- Pointwise resampling misses the long-run variance of an AR(1) mean Counterexample
- d-dimensional Brownian motion Definition
- Finite-dependent functional moving average Example
- Functions of disjoint independent coordinate blocks remain independent Example
- Gaussian AR(1) chain Example
- IID sequences as Markov chains with state-independent kernel Example
- Partial sums of independent centered variables are a martingale Example
- Product martingale from independent mean one factors Example
- Radial second moment of multidimensional Brownian motion Example
- Random-mapping representation for a finite transition matrix Example
- Simple random-walk transition kernel Example
- Square of a martingale minus quadratic compensator Example
- A nonconstant centered integrable IID random walk is unbounded above Lemma
- Bounded centered convergent series have summable variances Lemma
- Exact block budget for the covariance geometric-median pilot Lemma
- Independent-copy symmetrization of random series Lemma
- Jointly Gaussian blocks with zero cross-covariance are independent, including singular blocks Lemma
- Levy maximal bound from uniform tail bounds Lemma
- Long-run covariance of an L2-m-approximable shift is trace class Lemma
- One-sided maximal inequality for symmetric independent sums Lemma
- Sieve metric entropy produces uniform exponential tests Lemma
- Tail events are independent of every finite initial sigma-algebra Lemma
- The standard Wishart inverse quadratic form has an exact Fisher law Lemma
- Weighted overlapping-block moment and covariance limits Lemma
- Benjamini–Hochberg FDR under independence Theorem
- Exact t and partial F inference in full-rank Gaussian regression Theorem
- Finite-variance Hilbert mean bound for geometric medians of means Theorem
- Future-path Markov property Theorem
- Hoeffding inequality for independent bounded sums Theorem
- Kolmogorov maximal inequality Theorem
- Markov property of Brownian motion Theorem
- Orthogonal projections split a standard Gaussian into independent chi-square sums Theorem
- QR decomposition gives exact t and F pivots in the fixed Gaussian linear model Theorem
- Stratified estimator unbiasedness, variance, and proportional allocation Theorem
- The Gaussian sample mean is independent of the Wishart sample scatter Theorem
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
- Rick Durrett, Probability: Theory and Examples, 5th ed., Theorem 2.1.9 (standard reference, not scraped)