Continuous and discrete properties of stochastic processesTools Lee, Wai Ha (2010) Continuous and discrete properties of stochastic processes. PhD thesis, University of Nottingham.
AbstractThis thesis considers the interplay between the continuous and discrete properties of random stochastic processes. It is shown that the special cases of the onesided Lévystable distributions can be connected to the class of discretestable distributions through a doublystochastic Poisson transform. This facilitates the creation of a onesided stable process for which the Nfold statistics can be factorised explicitly. The evolution of the probability density functions is found through a FokkerPlanck style equation which is of the integrodifferential type and contains nonlocal effects which are different for those postulated for a symmetricstable process, or indeed the Gaussian process. Using the same Poisson transform interrelationship, an exact method for generating discretestable variates is found. It has already been shown that discretestable distributions occur in the crossing statistics of continuous processes whose autocorrelation exhibits fractal properties. The statistical properties of a nonlinear filter analogue of a phasescreen 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 interevent density of the process are found to be accurately approximated by a function of the Fano factor and the mean of the crossings alone.
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