High-dimensional Probability

High dimensional probability is the study of random vectors in R^n where the dimension n is large.

This modern research area is important in various applications such as

- Large random structures: random matrices, random networks.

- Statistics and machine learning

- Statistical physics

- Randomized algorithms

- Mixing times and other phenomena in high-dimensional Markov chains

An important aspect of this theory is non-asymptotic probabilistic bounds (concentration inequalities), i.e., inequalities which are or dimension free or where the role of the dimension is explicit.

The course will mainly focus on various concentration inequalities and applications.

Attention will also be given to Markov semigroups and interpolation methods based on them.

Concrete subjects treated in the course are

- Azuma Hoeffding inequality for martingale differences

- Mc Diarmid inequality, bounded difference inequality

- Sub-Gaussian random variables

- Sub-exponential random variables

- Efron-Stein inequality and tensorization of variance

- Poincare inequality and the Ornstein Uhlenbeck semigroup

- Log-Sobolev inequality and the Orstein Uhlenbeck semigroup

- Tensorization of entropy