Gaussian random vectors

2005/2006, sem. 2

Prof. Boris Tsirelson (School of Mathematical Sciences).
Time and place
Thursday 16-19 Schreiber 210.
Be acquainted with such things as the Hilbert space L2 of square integrable functions on a measure space. Everything else will be explained from scratch. However, some maturity in analysis is needed. (Maturity in probability is not needed.)

Note: no class on May 11 (because of the students day) and May 25 (because of the IMU conference).
Instead we plan class on May 8 and June 5 (13-16, Schreiber 210).

Syllabus (tentative)

Gaussian random vectors: equivalent definitions (in finite and infinite dimension).

Gaussian random trigonometric polynomials:
expected number of level crossings and extrema; expected Euler characteristics; the maximal value distribution.

Gaussian random fields on a sphere and a torus.

Regularity of stationary Gaussian processes: equivalent definitions and criteria.

Two quotes:

"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.

"Gaussian processes have a rich, detailed and very well understood general theory, which makes them beloved by theoreticians.
In applications [...] it is important to have specific, explicit formulae that allow one to predict, to compare theory with experiment, etc. As we shall see [...] it will be only for Gaussian (and related [...]) fields that it is possible to derive such formulae in the setting of excursion sets."

Adler and Taylor (the book cited below, p. 1).

Some literature

Lecture notes

  1. Basic notions: finite dimension.
    PDF or Postscript.
  2. Basic notions: infinite dimension.
    PDF or Postscript.
  3. Level crossings.
    PDF or Postscript.
  4. Extrema.
    PDF or Postscript.
  5. Random functions of two variables.
    PDF or Postscript.