Origin of Gaussian random variables

• Tools for the more formal explanation:

1. The PDF of the sum of random variables (X + Y) is the convolution of $f_X$ and $f_y$ ($f_X * f_y$, which is

$$\left( f_X * f_Y \right)(x) = \int\limits_{-\infty}^{\infty} f_X(\tau) f_Y(x - \tau) \;d\tau$$

2. The Fourier transform will help us transform convolution into multiplication.

$$\mathcal{F}\left\{ f_X * \cdots f_X \right\} = \mathcal{F}\left\{ f_X \right\} \cdots \mathcal{F}\left\{ f_X \right\}\; .$$

3. There is a particular sense of convergence, which will be explored later on, for which the expression $the\;distribution\;of\;the\;sum\;\;X_1 + \cdot X_n\;\;converges\;to\;a\;Gaussian\;distribution$.