Construction and Approximation of Stable Lévy Motion with Values in Skorohod Space

FieldValue
dc.contributor.authorSaidani, Becem
dc.date.accessioned2019-08-12T15:06:31Z
dc.date.available2019-08-12T15:06:31Z
dc.date.issued2019-08-12
dc.identifier.urihttp://hdl.handle.net/10393/39505
dc.identifier.urihttp://dx.doi.org/10.20381/ruor-23749
dc.description.abstractUnder an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable L´evy process. In this thesis, we developed an explicit construction for the α-stable L´evy process motion with values in D([0, 1]), by considering the cases α < 1 and α > 1. The case α < 1 is the simplest since we can work with the uniform topology of the sup-norm on D([0, 1]) and the construction follows more or less by classical techniques. The case α > 1 required more work. In particular, we encountered two problems : one was related to the construction of a modification of this process (for all time), which is right-continuous and has left-limit with respect to the J1 topology. This problem was solved by using the Itob-Nisio theorem. The other problem was more difficult and we only managed to solve it by developing a criterion for tightness of probability measures on the space of cadlag fonction on [0, T] with values in D([0, 1]), equipped with a generalization of Skorohod’s J1 topology. In parallel with the construction of the infinite-dimensional process Z, we focus on the functional extension of Roueff and Soulier [29]. This part of the thesis was completed using the method of point process, which gave the convergence of the truncated sum. The case α > 1 required more work due to the presence of centering. For this case, we developed an ad-hoc result regarding the continuity of the addition for functions on [0, T] with values in D([0, 1]), which was tailored for our problem.
dc.language.isoen
dc.publisherUniversité d'Ottawa / University of Ottawa
dc.subjectSkorohod space
dc.subjectStochastic process
dc.subjectProbability
dc.subjectFunctional limit theorem
dc.titleConstruction and Approximation of Stable Lévy Motion with Values in Skorohod Space
dc.typeThesis
dc.contributor.supervisorBalan, Raluca Madalina
thesis.degree.namePhD
thesis.degree.levelDoctoral
thesis.degree.disciplineSciences / Science
uottawa.departmentMathématiques et statistique / Mathematics and Statistics
CollectionThèses, 2011 - // Theses, 2011 -

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