**(Spring 2001)**

**R.Vershynin**

**Description of the course:**

Gaussian processes will be the basic object in this course. Starting with classical comparison theorems, our work will culminate in majorizing measures (nothing to do with measure theory!), a glass through which we look now at all bounded Gaussian processes. Applications will bring us to the convex geometry, clarifying also an extremely difficult result of Bourgain in Harmonic Analysis (Lambda-p problem).

I expect that some of our meetings will be in a seminar form, where somebody will present a topic, along with all the difficulties which we will try to resolve, etc.

**Prerequisites:**

standard probability and functional analysis.

There will be 12 lectures, one per week (on Thursdays
16:15 - 18:15 room 261),

starting March 15 and finishing June 7 (excluding the
Independence day, April 26).

I will keep posting readable notes for each lecture. Each
set of notes

contains the (most important) notions and results, and
also **all **exercises for

the current lecture. Solving the exercises is not required
but highly desirable

to develop intuition.

**Independent Random Variables**
**Lecture 1** (March 15)
**Lecture 2** (March 22)
**Lecture 3** (March 29)
**Lecture 4** (April 5)

No lecture (April 12) - Pesach
**Lecture 5** (April 19)

**Random Processes**
**Lecture 6** (May 3)
**Lecture 7** (May 10)
**Lecture 8** (May 17)
**Lecture 9** (May 24)
**Lecture 10** (May 24)

**FINAL HOMEWORK**: Due July
11, 2001

**Literature:**

M. Talagrand, Majorizing measures: the generic chaining.
Ann. Probab. 24 (1996), no. 3, 1049--1103

J.-P. Kahane, Some random series of functions. Library
codes 519.2 KAH, 519.28 KAH

M. Ledoux, M. Talagrand. Probability in Banach spaces.
Library code 519.2-LED