Continuous and discrete properties of stochastic processes - Lee, Wai Ha (2010)
This thesis considers the interplay between the continuous and discrete properties of random stochastic processes. It is shown that the special cases of the one-sided Lévy-stable distributions can be connected to the class of discrete-stable distributions through a doubly-stochastic Poisson transform. This facilitates the creation of a one-sided stable process for which the N-fold statistics can be factorised explicitly. The evolution of the probability density functions is found through a Fokker-Planck style equation which is of the integro-differential type and contains non-local effects which are different for those postulated for a symmetric-stable process, or indeed the Gaussian process. Using the same Poisson transform interrelationship, an exact method for generating discrete-stable variates is found. It has already been shown that discrete-stable distributions occur in the crossing statistics of continuous processes whose autocorrelation exhibits fractal properties. The statistical properties of a nonlinear filter analogue of a phase-screen model are calculated, and the level crossings of the intensity analysed. It is found that rather than being Poisson, the distribution of the number of crossings over a long integration time is either binomial or negative binomial, depending solely on the Fano factor. The asymptotic properties of the inter-event density of the process are found to be accurately approximated by a function of the Fano factor and the mean of the crossings alone.