Multivariate central limit theorem
Theorem taught in Estimation and inference for the multivariate normal, part of Statistical learning
Multivariate central limit theorem
| Type | Theorem |
| Taught in | Estimation and inference for the multivariate normal |
| Part | Statistical learning |
Let \mathbf{X}_1,\ldots,\mathbf{X}_n be independent and identically distributed p-dimensional random vectors with finite mean \boldsymbol{\mu} and covariance matrix \Sigma, and let \overline{\mathbf{X}}_n=n^{-1}(\mathbf{X}_1+\cdots+\mathbf{X}_n). Then
\sqrt{n}\,(\overline{\mathbf{X}}_n-\boldsymbol{\mu})
converges in distribution to \operatorname{N}_p(\mathbf{0},\Sigma) as n\to\infty.
Used in
- Taught in Estimation and inference for the multivariate normal, where the derivation and examples are.
- Also used in Chi-squared tests
- All 9 questions on Estimation and inference for the multivariate normal
See also
Central limit theorem · Multinomial central limit theorem · Mean vector and covariance matrix · Modes of convergence · The limit of Hotelling’s statistic · Invariance of the maximum likelihood estimator · Wishart distribution · Sampling law of the normal mean and scatter