In the previous post about limit theorems of stochastic processes, we considered when everything goes *right*, leading to Gaussian-like behavior. Here we’ll discuss when things go *wrong*, particularly when memory effects and infinite moments are introduced.

One way to understand the structure of randomness is to experience *a lot* of it. We’ll use $\lim_{n\to\infty} X_1 + \cdots + X_n$ as a case study, and along the way bump into classical ideas like the Central Limit Theorem (CLT) and Brownian motion.