Generalized Bernoulli process and fractional binomial distribution II

Jeonghwa Lee

Published 2022-09-04Version 1

Bernoulli process is a sequence of independent binary random variables, and binomial distribution concerns probabilities regarding its cumulative sum. Recently, a generalized Bernoulli process(GBP-I) was developed in which binary random variables are correlated with its covariance function obeying power law. The resulting sum of a finite sequence in GBP-I, called fractional binomial random variable, has variance asymptotically proportional to a fractional power of the length of the sequence. In this paper, we propose a different generalized Bernoulli process(GBP-II). It turns out that there is an interesting connection between GBP-II and fractional Poisson process. Similarities and differences between GBP-I, GBP-II, and fractional Poisson process are examined theoretically and empirically with simulations.