Let $p_n$ be some fixed pulse, for example $p_n =e^{-n^{2}}$
We have an infinite sum $y = \sum_{n=-\infty}^{\infty} a_n p_{-n}$ where $a_n$ are iid bernoulli random variables taking the values $+/- \sqrt{k}$ with equal probability.
Using the CLT, show this is approximately gaussian.
I'm not seeing how to correctly apply the CLT here - I can simulate the sum of many iterations in matlab and verify it looks gaussian, but I'm not sure how to show this.