Generalizations of Markov Chain Discretizations

Markov chains have famously been a crucial tool in understanding stochastic processes and queuing systems, among many other applications. Both discrete-time chains and continuoustime chains have been important centers in both research and application. These two cases are described by transition matrices. Continuous-time chains are difficult to model because this matrix is rather hard to compute in general. One attack to this problem is approximating a continuous-time chain with one that evolves in discrete time. The transition matrix is still difficult to compute exactly but can also be approximated to any order. The first-order approximation of this quantity is well-known. In 2008, Rachel Irby studied the second-order approximation and compared its performance to that of the first-order counterpart. In this paper, we will generalize the results outlined in Rachel\u27s paper by using an approximation of any order. Specifically, we will study direct comparisons using norms. In addition, we will compare the continuous-time chain to its approximate model through the study of stationary and limiting distributions