Origin of Gaussian random variables
- Tools for the more formal explanation:
- 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
\[ \left( f_X * f_Y \right)(x) = \int\limits_{-\infty}^{\infty} f_X(\tau) f_Y(x - \tau) \;d\tau \]
- The Fourier transform will help us transform convolution into multiplication.
\[ \mathcal{F}\left\{ f_X * \cdots f_X \right\} = \mathcal{F}\left\{ f_X \right\} \cdots \mathcal{F}\left\{ f_X \right\}\; . \]
- There is a particular sense of convergence, which will be explored later on, for which the expression \(the\;distribution\;of\;the\;sum\;\;X_1 + \cdot X_n\;\;converges\;to\;a\;Gaussian\;distribution\).