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 beendone when the AR(1) process is built upon extreme value random variables. This processappears when the distribution of the current maximum is dependent on the previous. Thegoal of this dissertation is to explore asymptotic expansions of tail probabilities on thistopic, in particular using the Gumbel distribution. In each of the theoretical projects webuild second-order expansions, many of which are improvements over the already knownfirst-order ones. We also examine exactly when each of the expansions should and shouldnot be used through simulation studies. Finally, we perform a data analysis in the extremevalue theory setting on riverflow data, and as much as possible connect this same data set to the theoretical results.