1. On page 461 it is said that the convergence of the characteristic function implies the convergence in probability. To prove this one needs of course the independence of increments. The detailed reasoning is on pages 62-63.

2. Using the Lévy-Khintchine formula one can easily prove Proposition 1.114: If the jumps of a stochastically continuous process with independent increment are bounded then all the moments of the process are finite.Download