Doubly stochastic Poisson point process driven by fractal shot noise

We explore the behavior of the fractal-shot-noise-driven doubly stochastic Poisson point process (FSNDP), for which the associated impulse response functions assume a decaying power-law form. For a variety of parameters of the process, we obtain expressions for the count-number distribution and moments, Fano factor, normalized coincidence rate, power spectral density, and time probability densities. A number of these measures exhibit power-law dependencies, indicating fractal behavior. For certain parameters the power spectral density exhibits 1/f-type behavior over a substantial range of frequencies, so that the process serves as a 1/f^α point process for {α} in the range 02. We consider two physical processes that are well described by the FSNDP: Cherenkov radiation from a random stream of charged particles, and diffusion of randomly injected concentration packets.