First passage times in markov chains

First passage time in Markov chains is defined as the first time one chain?s passes a specified state or set of states. First passage time is a random variable depending on time. This state or states may indicate first passage time of an interesting, rare and amazing event . In this study, the distribution of the first passage time has been tried to be obtained for an irreducible Markov chain whose state space is finite. Also, the lumping method which aims at the reduction of the state space and which is done by bringing the states together has been explained. The distribution of the first passage time has been tried to be determined with lumping as well. Johnson SB distribution has come to the forefront in determining the distribution of the first passage time of irreducible Markov chain with four states considered. Depending on the trial number of the first passage time distribution, convergences have been tried to be shown by using Kullback distance measure and Wasserstein metric. Furthermore, transition matrices approaching to the desired limit distribution and entropy of limit distribution have been obtained using the delumping method.