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|Title: ||Stable and multistable processes and localisability|
|Authors: ||Liu, Lining|
|Supervisors: ||Falconer, K. J.|
|Issue Date: ||23-Jun-2010|
|Abstract: ||We first review recent work on stable and multistable random processes and their
localisability. Then most of the thesis concerns a new approach to these topics
based on characteristic functions.
Our aim is to construct processes on R, which are α(x)-multistable, where the
stability index α(x) varies with x. To do this we first use characteristic functions
to 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 the
limit of a sequence of measures made up of α-stable random measures restricted
to small intervals with α constant on each interval.
We then use the multistable random integrals to define multistable random
processes on R and study the localisability of these processes. Thus we find conditions
that ensure that a process locally ‘looks like’ a given stochastic process
under enlargement and appropriate scaling. We give many examples of multistable
random processes and examine their local forms.
Finally, we examine the dimensions of graphs of α-stable random functions
defined by series with α-stable random variables as coefficients.|
|Publisher: ||University of St Andrews|
|Appears in Collections:||Pure Mathematics Theses|
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