In 1990, Jakeman (see \cite{jakeman1990statistics}) defined the binomial process as a special case of the classical birth-death process, where the probability of birth is proportional to the difference between a fixed number and the number of individuals present. Later, a fractional generalization of the binomial process was studied by Cahoy and Polito (2012) (see \cite{cahoy2012fractional}) and called it as fractional binomial process (FBP). In this paper, we study second-order properties of the FBP and the long-range behavior of the FBP and its noise process. We also estimate the parameters of the FBP using the method of moments procedure. Finally, we present the simulated sample paths and its algorithm for the FBP.

翻译：1990年，Jakeman（见文献\cite{jakeman1990statistics}）将二项过程定义为经典生灭过程的一种特例，其中出生概率与固定数量与当前个体数量之差成正比。随后，Cahoy和Polito（2012）（见文献\cite{cahoy2012fractional}）研究了二项过程的分数推广形式，并将其称为分数二项过程（FBP）。本文研究了FBP的二阶性质及其噪声过程的长程行为，并运用矩估计法对FBP的参数进行了估计。最后，我们给出了FBP的模拟样本路径及其算法。