C,\delta) condition from the Talagrand's decomposition theorem for infinitely divisible processes

In the nineties, Talagrand proved that every general infinitely divisible process can be divided into two parts: one controlled by Bernstein's inequality and the other, which is a Levy positive process. The result works under a strong assumption on the Levy measure known as H (C, \delta) condition, which specifically excludes Dirac measures. Relaxing the assumption must be a challenge because it encounters similar difficulties as in the proof of Bernoulli's Theorem. In the lecture there will be described the new Talagrand's proof of his old result on infinitely divisible processes based on a certain corollary of Bernoulli's Theorem (previous proof was very complicated and vague). There will be also explained how to apply the Talagrand approach in such a way that it extends its result beyond the assumption of H (C, \delta).