A Martingale Approach to Expected Waiting Time in Higher Order Markov Chain

[[abstract]]In this thesis we mainly aim to compute the expected waiting time of a certain pattern in Markov chain of order $m$ and the relative probabilities of every ending scenarios. Moreover, we can calculate the closed form of the expected waiting and generalize the results in Glaz, Kulldorff, Pozdnyakov, and Steele (2006)