Recently there has been an interest in asymptotic expansions of the tail probabilities of a variety of processes that are ubiquitous in statistics. However, little to no work has been done when the AR(1) process is built upon extreme value random variables. This process appears when the distribution of the current maximum is dependent on the previous. The goal of this dissertation is to explore asymptotic expansions of tail probabilities on this topic, in particular using the Gumbel distribution. In each of the theoretical projects we build second-order expansions, many of which are improvements over the already known first-order ones. We also examine exactly when each of the expansions should and should not be used through simulation studies. Finally, we perform a data analysis in the extreme value theory setting on riverflow data, and as much as possible connect this same data set to the theoretical results.