In this appendix, we will proof some important results for the geometric proof of the channel capacity of the additive white Gaussian noise channel.

Let n = [n₁, n₂,…, n_D] be the noise vector composed of D independent terms. The probability density of each of these terms is Gaussian:

[A.1]

The probability density of the vector n is the product of the D probability densities p(n_i):

[A.2]

Let be the norm of n. Since the variance of n_i is = σ², the mean of r² is given by:

[A.3]

And the variance of is:

[A.4]

For high values of D, using the limit central theorem, we can show that the variance of r² is equal to 2σ⁴D. Consequently, when D tends to infinite, the norm r is concentrated around