I was going through a blog which details how to generate a multivariate Gaussian vector, given a mean vector μ and co-variance matrix σ.
As a starting point, author uses generated uniform random numbers to simulate Gaussian random numbers using central limit theorem. The relevant extract from the blog is below:
Let's say you generate m uniform random numbers (each between 0 and 1) and you use the variable xi to denote each of these. The Central Limit Theorem allows us to convert these m numbers belonging to U(0,1) into a single number that belongs to the Guassian distribution N(0,1).
$$x = \frac{\sum_ix_i-\frac{m}{2}}{\sqrt{\frac{m}{12}}}$$
Here, x is a one dimensional Gaussian random number – produced using the help of m uniform random variables. The $\frac{m}{2}$ is derived from the term mμu (where μu is the mean of the uniform distribution – $\frac{1-0}{2} = 0.5$). The denominator is derived from the term σ$\sqrt{m}$ where $\sigma^2$ is the variance of the uniform distribution between 0 and 1 (comes to exactly $\frac{1}{12}$).
Link to the article : http://www.aishack.in/tutorials/generating-multivariate-gaussian-random/
I wanted to understand the details of the underlying math as I am not sure how central limit theorem is used to generate N(0,1) number from m U(0,1) numbers. I googled the same but was not able to find an explanation for this.
Best Answer
The idea is that the if there are $m$ independent random variables $X_i$ each uniformly distributed on $[0,1]$, so with mean $\frac12$ and variance $\frac1{12}$, then their sum $S_m$ has mean $\frac{m}2$ and variance $\frac{m}{12}$ while their average $\frac1m S_m$ has mean $\frac{1}2$ and variance $\frac{1}{12m}$
The Central Limit theorem says that $\sqrt{m}\left(\frac1m S_m - \frac{1}2\right)$ converges in distribution to $\mathcal N\left(0,\frac1{12}\right)$, i.e. $\dfrac{ S_m - \frac{m}2}{\sqrt{\frac{m}{12}}}$ converges in distribution to $\mathcal N(0,1)$ as $m$ increases
So for sufficiently large $m$, you might use $\dfrac{ S_m - \frac{m}2}{\sqrt{\frac{m}{12}}}$ to generate a random variable which has a distribution close to that of a standard normal random variable