2.2. Definition and characterization of Gaussian vectors
DEFINITION.– We say that a real random vector XT = (X1,…, Xn) is Gaussian if ∀(a0, a1,…, an) ∈ 
n+1 the r.v. 
is Gaussian (we can in this definition assume that a0 = 0 and this will be sufficient in general).
A random vector XT = (X1,…, Xn) is thus not Gaussian if we can find an n -tuple (a1,…, an) ≠ (0,…, 0) such that the r.v. 
is not Gaussian and for this it suffices to find an n -tuple such that 
is not an r.v. of density.
EXAMPLE.– We allow ourselves an r.v. X ∼ N (0,1) and a discrete r.v. 
, independent of X and such that:
We state that Y = X .
By using what has already been discussed, we will show through an exercise that although Y is an r.v. N (0, 1), the vector (X, Y) is not a Gaussian vector.
PROPOSITION.– In order for a random ...
Get Discrete Stochastic Processes and Optimal Filtering now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
