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Asymptotic analysis of some point processes

Abstract : Stein’s method constitutes one of the main techniques to solve some approximation problems in probability theory. In this manuscript, we apply it in the context of point processes. The first part of these investigations focuses on the Poisson point process. Its characteristic independence property provides a way to explain intuitively why a sequence of point processes becoming less and less repulsive can converge to such a point process. More generally, this leads to show some convergence results for some sequences of point processes built by several operations such as superposition, thinning and rescaling. The use of a distance on point processes, the so-called Kantorovich-Rubinstein distance, enables moreover the getting of some convergence rates. The second part is centered on a class of point processes with important attractiveness, called discrete α-stable point processes. Their structure based on a Poisson point process gives us a way to enlarge to these point processes the method used previously and to propose new results, via some properties that we state on these point processes.
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Aurélien Vasseur. Asymptotic analysis of some point processes. Probability [math.PR]. Télécom ParisTech, 2017. English. ⟨NNT : 2017ENST0062⟩. ⟨tel-03413164⟩

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