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. |