Fractional Poisson Process

For almost two centuries, Poisson process with memoryless property of corresponding exponential distribution served as the simplest, and yet one of the most important stochastic models. On the other hand, there are many processes that exhibit long memory (e.g., network traffic and other complex systems). It would be useful if one could generalize the standard Poisson process to include these processes. This generalization adds a parameter $alin (0, 1]$, and is called the fractional exponent of the process. In this thesis, we clearly derive the transition from standard Poisson process to its fractional generalization (fractional Poisson process (fPp)). The link fPp and $alpha$-stable density is established by solving an integral equation. The link then leads to an algorithm for generating fPp that discovering more interesting properties. Method-of-moments estimators for the intensity rate $mu$ and fractional order $alpha$ derived and showing asymptotic normality of the estimators and construction of the corresponding confidence interval. Then the properties of the estimators are then tested using simulated data.