Abstract
In this paper, we present a Bernoulli difference Markov model and a Bernoulli difference time series model based on Jacobs-Lewis mixture method. The limiting distribution of Jacobs-Lewis mixture model is obtained. The Bernoulli difference Markov model allows for positive and negative correlation. Maximum likelihood, conditional maximum likelihood, and Yule Walker methods of estimation are considered. Simulations are carried out. The paper concludes with an analysis of a real data set.