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Covariance is symmetric and bilinear in finite linear combinations
Statement
Let and be finite families of square-integrable real random variables on one common probability space. Then covariance is symmetric, and for real scalars , This includes empty sums and constant summands. On a finite probability space all real random variables are square-integrable, so the original finite-space case is included with the same centered-product covariance.
Facts & Assumptions
Given: The square-integrable families on one common probability space and real scalars in the Statement.
Expectation is linear on finite linear combinations (Linearity, monotonicity, and the modulus bound for expectation).
Proof
Each variable is integrable because on a probability space, and every product of two such variables is integrable because . Finite real linear combinations remain square-integrable by repeated use of . Thus all covariances and expectations below are defined and finite. Commutativity of real multiplication in [L2] gives .
The finite linear combinations and their product have the required integrability by step 1.1. Substitute them into [L2] and distribute their pointwise product.
Applying [L1] to step 2.1 and collecting the terms gives the displayed double sum.
If either index set is empty, both sides are zero; covariance with a constant is zero by [L1] and [L2].
Depends on
Used by
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- Unequal error variances can make OLS fail to be BLUE Counterexample
- A deterministic integral construction of a Gaussian process Example
- Brownian bridge from Brownian motion Example
- Classical and Berkson slopes, misclassification, and validation Example
- Covariance of overlapping Brownian increments Example
- Linear combinations of Brownian values are Gaussian Example
- Perfect correlation can coexist with a method offset Example
- Repeated method differences can share a random intercept Example
- A Gaussian error contrast has the induced multivariate-normal law Lemma
- Continuous and strictly increasing norm distributions for density bridges Lemma
- Geometric strong mixing gives an Lp Poisson solution Lemma
- Marginal covariance of the linear mixed model Lemma
- The observed law need not identify the latent-error variance split Proposition
- Antithetic variance and a monotone sufficient condition Theorem
- Berkson error preserves the simple linear regression slope Theorem
- Classical error attenuates the simple linear regression slope Theorem
- Conditional Gaussian law of random effects Theorem
- Design expectation and variance of a paired contrast Theorem
- Equal-size cluster design effect under exchangeable within-cluster covariance Theorem
- First-order propagation of measurement uncertainty Theorem
- Full-rank OLS is unbiased with its classical covariance Theorem
- Gauss–Markov theorem for estimable contrasts Theorem
- Horvitz–Thompson variance and pair-inclusion estimate Theorem
- Known-covariance BLUP and its prediction-error covariance Theorem
- Known-covariance GLS is BLUE Theorem
- Mean and variance under balanced biological and technical replication Theorem
- Unbiasedness and optimal coefficients for fixed controls Theorem
- Unbiasedness and variance decomposition for the two-stage expansion total Theorem
- Unbiasedness and variance of the SRSWOR cluster-total estimator Theorem
- Variance of a finite sum as the sum of all variances and covariances Theorem
Dependency tree · two levels
12 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
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 5.3.1 (standard reference, not scraped)