Generalized Bernoulli Process and Fractional Binomial Distribution

Abstract

Recently, a generalized Bernoulli process (GBP) was developed as a stationary binary sequence whose covariance function obeys a power law. In this paper, we further develop generalized Bernoulli processes, reveal their asymptotic behaviors, and find applications. We show that a GBP can have the same scaling limit as the fractional Poisson process. Considering that the Poisson process approximates the Bernoulli process under certain conditions, the connection we found between GBP and the fractional Poisson process is thought of as its counterpart under long-range dependence. When applied to indicator data, a GBP outperforms a Markov chain in the presence of long-range dependence. Fractional binomial models are defined as the sum in GBPs, and it is shown that when applied to count data with excess zeros, a fractional binomial model outperforms zero-inflated models that are used extensively for such data in current literature.