Continuous and discrete properties of stochastic processes
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 transfor...
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| Format: | Thesis (University of Nottingham only) |
| Language: | English |
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2010
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| Online Access: | https://eprints.nottingham.ac.uk/11194/ |
| _version_ | 1848791216932519936 |
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| author | Lee, Wai Ha |
| author_facet | Lee, Wai Ha |
| author_sort | Lee, Wai Ha |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T18:24:59Z |
| format | Thesis (University of Nottingham only) |
| id | nottingham-11194 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| language | English |
| last_indexed | 2025-11-14T18:24:59Z |
| publishDate | 2010 |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-111942025-02-28T11:11:54Z https://eprints.nottingham.ac.uk/11194/ Continuous and discrete properties of stochastic processes Lee, Wai Ha 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. 2010-07-19 Thesis (University of Nottingham only) NonPeerReviewed application/pdf en arr https://eprints.nottingham.ac.uk/11194/1/Thesis_Wai_Ha_Lee.pdf Lee, Wai Ha (2010) Continuous and discrete properties of stochastic processes. PhD thesis, University of Nottingham. continuous stable distribution Gaussian distribution discrete stable distribution Poisson distribution Fano factor fracal properties closed-form stable distributions doubly stochastic Poisson transform doubly stochastic Gaussian transform Fokker-Planck equation phase screen model binomial negative binomial crossing statistics inter-event density persistence |
| spellingShingle | continuous stable distribution Gaussian distribution discrete stable distribution Poisson distribution Fano factor fracal properties closed-form stable distributions doubly stochastic Poisson transform doubly stochastic Gaussian transform Fokker-Planck equation phase screen model binomial negative binomial crossing statistics inter-event density persistence Lee, Wai Ha Continuous and discrete properties of stochastic processes |
| title | Continuous and discrete properties of stochastic processes |
| title_full | Continuous and discrete properties of stochastic processes |
| title_fullStr | Continuous and discrete properties of stochastic processes |
| title_full_unstemmed | Continuous and discrete properties of stochastic processes |
| title_short | Continuous and discrete properties of stochastic processes |
| title_sort | continuous and discrete properties of stochastic processes |
| topic | continuous stable distribution Gaussian distribution discrete stable distribution Poisson distribution Fano factor fracal properties closed-form stable distributions doubly stochastic Poisson transform doubly stochastic Gaussian transform Fokker-Planck equation phase screen model binomial negative binomial crossing statistics inter-event density persistence |
| url | https://eprints.nottingham.ac.uk/11194/ |