Wai Ha Somebody searched for "negative binomial as 'doubly stochastic poisson'" on Google yesterday. One excellent reference on doubly stochastic distributions is the paper by M. C. Teich and P. Diament, "Multiply stochastic representations for K distribu... more
Wai Ha Somebody searched for "ten discrete and ten continuos [sic] distrubutions" yesterday. In the 'talk' part of the stable distributions page on Wikipedia (link below), I have listed eleven closed-form expressions for (continuous) stable distributions.... more
Wai Ha (3/3) Discrete-stable distributions and processes arise, for example, as the stationary state of the Death, Multiple-Immigration (DMI) process. You can define a one-sided continuous-stable process through discrete-stable process; this is the basis ... more
Wai Ha Somebody searched for "ten discrete and ten continuos [sic] distrubutions" yesterday. In the 'talk' part of the stable distributions page on Wikipedia (link below), I have listed eleven closed-form ... more
Alumnus, Mathematics
Postgraduate Researcher
Thesis Title: Continuous and discrete properties of stochastic processes
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Keith Hopcraft
Eric Jakeman |
About
I studied for a PhD in Mathematics in the theoretical mechanics division of the University of Nottingham's Mathematics department between 2005 and 2009.
My thesis (full text available in my 'websites' section) deals mainly in the class of 'stable distributions', which can be found in physical systems, the internet, social networks and even semantics. The main point of the third chapter, which is disseminated in my paper 'Continous and discrete stable processes', is the definition of a continuous-stable stochastic process through a 'doubly stochastic Poisson transform pair'.
Thesis abstract:
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.
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