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

Bernoulli process is a finite or infinite sequence of independent binary variables, X-i, i = 1, 2, ..., whose outcome is either 1 or 0 with probability P(X-i = 1) = p, P(X-i = 0) = 1 - p, for a fixed constant p is an element of (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 2H - 2, H is an element of (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 n consecutive variables in a generalized Bernoulli process, of particular interest is when its variance is proportional to n(2H), if H is an element of (1/2, 1).