2.1 Some reminders regarding random Gaussian vectors

DEFINITION. We say that a real r.v. is Gaussian, of expectation m and of variance ? ² if its law of probability P _X:

• admits the density f _X( x) if ? ² ? 0 (using a double integral calculation, for example, we can verify that f _X( x) dx = 1);

• is the Dirac measure ? _m if ? ² = 0.

Figure 2.1: Gaussian density and Dirac measure

If ? ² ? 0, we say that X is a non-degenerate Gaussian r.v.

If ? ² = 0, we say that X is a degenerate Gaussian r.v.; X is in this case a "certain r.v." taking the value m with the probability 1.

EX = m, Var X = ? ². This can be verified easily by using the probability distribution function.

As we have already observed, in order to specify that an r.v. X is Gaussian of m expectation and of ? ² variance, we will write X ? N ( m, ? ²).

Characteristic function of X ? N ( m, ? ²)

Let us begin firstly by determining the characteristic function of X ₀ ? N(0,1):

We can easily see that the theorem of derivation under the sum sign can be applied:

Following this by integration by...