Multivariate normal distribution

A random vector X=(X₁,...,X_n) follows a multivariate normal distribution, also sometimes called a multivariate Gaussian distribtuion (in honor of Carl Friedrich Gauss, who was not the first to write about the normal distribution), if it satisfies the following equivalent conditions:

• every linear combination Y=a₁X₁+...+a_nX_n is normally distributed;
• there is a random vector Z=(Z₁,...,Z_m), whose components are independent standard normal random variables, a vector μ=(μ₁,...,μ_n) and an n-by-m matrix A such that X=A Z + μ.
• there is a vector μ=(μ₁,...,μ_n) and a symmetric, positive semidefinite matrix Γ such that X has density

f_X(x₁,...,x_n)dx₁...dx_n = (det(2πΓ))^-1/2 exp ½((X-μ)^TΓ^-1(X-μ)) dx₁...dx_n

• there is a vector μ and a symmetric, positive definite matrix Γ such that the characteristic function of X is

φ_X(u)=exp(iμ^Tu-½u^TΓu).

The vector μ in these conditions is the expected value of X and the matrix Γ=A^TA is the covariance matrix of the components X_i. It is important to realize that the covariance matrix must be allowed to be singular. That case arises frequently in statistics; for example, in the distribution of the vector of residuals in ordinary linear regression problems. Note also that the X_i are in general not independent; they can be seen as the result of applying the linear transformation A to a collection of independent Gaussian variables Z.

Proof?

Multivariate Gaussian density

Recall characteristic function of a random vector.

Recall characterizations of gaussian random variables.

Calculate characteristic function of Z in terms of characteristic function of X.

Deduce characteristic functional of X in terms of mean vector and covariance matrix.