TAU:0366-4765

Gaussian measures

2010/2011, sem. 1

Lecturer
Prof. Boris Tsirelson (School of Mathematical Sciences).
Prerequisites
Be acquainted with such things as the Hilbert space L2 of square integrable functions on a measure space, and the normal distribution. Everything else will be explained from scratch. However, some maturity in analysis is needed. (Maturity in probability is not needed.)
Grading policy
Written homework and oral exam.

A quote:

"Gaussian random variables and processes always played a central role in the probability theory and statistics. The modern theory of Gaussian measures combines methods from probability theory, analysis, geometry and topology and is closely connected with diverse applications in functional analysis, statistical physics, quantum field theory, financial mathematics and other areas."

R. Latala, On some inequalities for Gaussian measures. Proceedings of the International Congress of Mathematicians (2002), 813-822. arXiv:math.PR/0304343.

Lecture notes: results formulated

All these results will be discussed, but only some of them will be proved.
  1. Functions of normal random variables.
  2. Random real zeroes.
  3. Sensitivity and superconcentration.

Lecture notes: proofs and more

  1. Random real zeroes: no derivatives.
  2. Random real zeroes: one derivative.
  3. Random real zeroes: two derivatives.
  4. Sensitivity and superconcentration.

Lecture notes: infinite dimension

  1. Gaussian spaces and processes.
  2. Functions of normal random variables.
  3. Random real zeroes: no derivatives.
  4. Random real zeroes: one derivative.