Stable and multistable processes and localisability

Stochastic processes;Characteristic functions

Liu, Lining ; Falconer, K. J. ; Falconer, K. J.

University:University of St Andrews

Department:Mathematics & Statistics (School of)

关键词: Stochastic processes;    Characteristic functions;   

Others  :  https://research-repository.st-andrews.ac.uk/bitstream/handle/10023/948/Lining%20Liu%20PhD%20thesis.PDF?sequence=3&isAllowed=y

来源: DR-NTU

【 摘 要 】

We first review recent work on stable and multistable random processes and theirlocalisability. Then most of the thesis concerns a new approach to these topicsbased on characteristic functions.Our aim is to construct processes on R, which are α(x)-multistable, where thestability index α(x) varies with x. To do this we first use characteristic functionsto define α(x)-multistable random integrals and measures and examine their properties.We show that an α(x)-multistable random measure may be obtained as thelimit of a sequence of measures made up of α-stable random measures restrictedto small intervals with α constant on each interval.We then use the multistable random integrals to define multistable randomprocesses on R and study the localisability of these processes. Thus we find conditionsthat ensure that a process locally ‘looks like’ a given stochastic processunder enlargement and appropriate scaling. We give many examples of multistablerandom processes and examine their local forms.Finally, we examine the dimensions of graphs of α-stable random functionsdefined by series with α-stable random variables as coefficients.