Fractional Poisson process in terms of alpha -stable densities

The link between fractional Poisson process (fPp) and α-stable density is established by solving an integral equation. The result is then used to study the properties of fPp such as asymptotical n-th arrival time, number of events distributions, covariance structure, stationarity and dependence of increments, self-similarity, and intermittency property. Asymptotically normal parameter estimators and their variants are derived; their properties are studied and compared using synthetic data. An alternative fPp model is also proposed. Finally, the asymptotic distribution of a scaled fPp random variable is shown to be free of some parameters; formulae for integer-order, non-central moments are also derived.

Keywords. fractional Poisson process, α-stable, intermittency, scaled fPp, self-similarity