Normal Distribution Properties
Property 1: If X and A are k × 1 column vectors and X ∼ N(μ, Σ), then
Definition 1: The standard multivariate normal distribution is a multivariate normal distribution where the mean vector μ is the zero vector and the covariance matrix is the identity matrix.
To standardize a vector X, to obtain the standardized vector Z
Z = (UT)-1(X – μ)
you have two choices for U: (1) use the Cholesky decomposition Σ = UTU or (2) use the square root matrix U = Σ1/2 as described in Positive Definite Matrices. These choices exist since Σ is a positive definite matrix.
Property 2: X ∼ N(μ, Σ), then the standard vector Z ∼ N(0, I).
Property 3:
Property 4: Suppose X and Y are independent k × 1 column vectors. If X ∼ N(μX, ΣX) and Y ∼ N(μY, ΣY), then
Property 5: If X1, …, Xn ∼ N(μ, Σ) is a random sample then
Property 6 (Multivariate Central Limit Theorem): If X1, …, Xn is a random sample from a population with mean μ and covariance matrix Σ, then for n sufficiently large
This means that as n → ∞, the distribution of
See also Multivariate Central Limit Theorem.
Wishart Distribution
If X1, …, Xn ∼ N(0, Σ) are independent, then X12 + … + Xn2 is said to have a Wishart distribution with n degrees of freedom, denoted W(n, Σ).
Note that this is the multivariate counterpart to the chi-square distribution since if x1, …, xn ∼ N(0, 1) are independent, then x12 + … + xn2 ∼ χ2(n). We also know that (n–1)s2/σ2 ∼ χ2(n–1). The multivariate counterpart is
Property 7: If X1, …, Xn ∼ N(0, Σ) is a random sample, then
where
References
Rencher, A.C. (2002) Methods of multivariate analysis (2nd Ed). Wiley-Interscience, New York.
http://ndl.ethernet.edu.et/bitstream/123456789/27185/1/Alvin%20C.%20Rencher_2012.pdf
Azevedo, C. L. N. (2016) The Multivariate Normal Distribution
https://www.ime.unicamp.br/~cnaber/mvnprop.pdf