The thesis investigates a pair of stationary stochastic process models whose domains are the set of integers and the set of real numbers respectively. The stationary processes with our specific correlation functions include the discrete and continuous first and second order autoregressive processes as their special cases. The maximum likelihood method is then applied to obtain the nonlinear equation system for the maximum likelihood estimators of the model parameters and the solutions are found by using the deepest gradient algorithm. The advantage of the algorithm lies in the calculation could be divided into several steps at a cost of O(n) calculations per step. Finally, predictions are given for both simulated data and Wichita temperature data.