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

Abstract

Under 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.