Research objectives and content
The contraction principle is one of the fundamental principles in the probability theory in Banach spaces. It consists in replacing a 'complicated' Banach space valued random variable by an 'easier' one where the corresponding absolute moments are comparable in at least one direction. This principle turned out to be powerful in several applications. Up to now the research has been concentrated on the situation that an unconditional sum of certain random variables represents the 'complicated' random variable whereas the 'easier' variable is a sum of independent Rademacher variables.
My objective of the project consists in replacing the unconditional sums by sums of martingale differences and in replacing the Rademacher functions by other relevant families of random variables, such as stable and exponential ones. For a special case, the Gaussian variables, I have already proved a contraction principle with the help of Talagrand's majorizing measure theorem for Gaussian processes.
Training content (objective, benefit and expected impact)
The University Paris VI is a leading center in the interaction between probability theory and functional analysis and provides strong support in both parts of the project, the stochastic process part (for example M. Talagrand) and the part concerning Banach space valued martingales (G. Godefroy, G. Pisier, Q. Xu).