Discrete Stochastic Processes and Optimal Filtering, 2nd Edition

Chapter 2

Gaussian Vectors

2.1. Some reminders regarding random Gaussian vectors

DEFINITION.– We say that a real r.v. is Gaussian, of expectation m and of variance σ² if its law of probability P_X:

– admits the density if σ² ≠ 0 (using a double integral calculation, for example, we can verify that ;

– is the Dirac measure δ_m if σ² = 0.

Figure 2.1. Gaussian density and Dirac measure

If σ² ≠ 0, we say that X is a non-degenerate Gaussian r.v.

If σ² = 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 = σ². 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 expectation m and of variance σ², we will write X ~ N(m,σ²).

Characteristic function of X ~ N(m,σ²)

Let us begin by first determining the characteristic function of X₀ ~ N(0,1):

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

Following this, with integration by parts:

The resolution of the differential ...