  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.
