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Covariance matrix of a finite-dimensional random vector
Definition
Let be a measurable random vector in , where is finite, and assume . Define the mean coordinatewise by and define the covariance matrix by where each entry is the expectation of the corresponding real random variable. Write .
The well-definedness lemma proves these expectations are finite, is symmetric positive semidefinite, and .
Depends on
Used by
- Adaptive truncated covariance estimator Algorithm
- Asymptotic normality of the Cox partial-likelihood estimator Corollary
- Intrinsic matrix variance alone does not control pilot-centered covariance truncation Counterexample
- Matrix variance need not dominate trace covariance times operator norm Counterexample
- Bounded-spectrum covariance class Definition
- Covariance and correlation principal component analysis Definition
- Gaussian Huber contamination model for covariance estimation Definition
- Matrix variance for covariance estimation Definition
- A polynomial-tail law with finite matrix variance Example
- Rescaling can change covariance-PCA directions Example
- A finite second moment defines a positive semidefinite covariance matrix Lemma
- Exact block budget for the covariance geometric-median pilot Lemma
- Oracle-truncated covariance and vector concentration under finite matrix variance Lemma
- Sample-centered covariance is a measurable statistic and estimator Lemma
- The covariance matrix variance ratio is at least one Lemma
- Finite-fourth-moment root-n consistency of sample-centered covariance Theorem
- IID terminal influence limit from a separately established expansion Theorem
- Independent-group terminal influence limit Theorem
- Terminal normal limit for weighted logrank scores Theorem
- The leading PCA eigenspace maximizes projected variance Theorem
Dependency tree · two levels
13 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
- Philippe Rigollet, 18.S997 High-Dimensional Statistics Lecture Notes, MIT OpenCourseWare, Chapter 4 (standard reference, not scraped)