Generalized Bernoulli process with long-range dependence and fractional binomial distribution

Bernoulli process is a finite or infinite sequence of independent binary variables, , = 1, 2, · · ·, whose outcome is either 1 or 0 with probability = 1) = , = 0) = 1 – , for a fixed constant ∈ (0, 1). We will relax the independence condition of Bernoulli variables, and develop a generalized Bernoulli process that is stationary and has auto-covariance function that obeys power law with exponent 2 – 2, ∈ (0, 1). Generalized Bernoulli process encompasses various forms of binary sequence from an independent binary sequence to a binary sequence that has long-range dependence. Fractional binomial random variable is defined as the sum of consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to , if ∈ (1/2, 1).