One of the primary activities in probability theory is studying the various statistics that can be formed from a complex system with many components. For instance, one of the simplest systems one can consider at simplest is a finite or infinite sequence \(X_1,\ ...\ ,\ X_n\ \) or \(X_1,\ X_2,\ ...\) of jointly independent scalar random variables with the case that the \(X_i\) are identically distributed (or the \(X_i\) are iid for short), being a model case of interest. There are an abundance of statistics that one can study with such sequences, but one of the most basic such statistics are the partial sums