Lecturer

Prof. Boris Tsirelson (School of Mathematical Sciences).

Instructor

Alon Nishry.

Prerequisites

Functions of a real variable; introduction to probability theory.

Grading policy

The final exam (24.06; 30.08).

LECTURE NOTES

• Part A: Independence
  • 1: Long independent sequences
  • 2: Central limit theorem
  • 3: Infinite independent sequences
• Part B: Stopping
  • 4: Random walks
  • 5: Markov chains
• Part C: Dependence
  • 6: Martingales
  • 7: Conditioning

TEXTBOOKS

• KORALOV Leonid, SINAI Yakov, "Theory of probability and random processes" (second edition, Springer, 2007).
• DURRETT Richard, "Probability: theory and examples" (second edition, Cornell/Wadsworth 1996).
• WILLIAMS David, "Probability with martingales" (Cambridge 1991-2008).
• POLLARD David, "A user's guide to measure theoretic probability" (Cambridge 2002).

EXERCISES (by Alon Nishry, in Hebrew)

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

EXAMS (in Hebrew)

• 2010, sample
• 14.06.2010
• 20.08.2010
• 2011, sample
• 13.06.2011
• 16.09.2011
• 24.06.2013
• 30.08.2013

Some quotes:

Probability theory has a right and a left hand. On the left is the rigorous foundational work using the tools of measure theory. The right hand `thinks probabilistically'.

DURRETT, page xii (after Breiman).

...measure theory, that most arid of subjects when done for its own sake, becomes amazingly more alive when used in probability, not only because it is then applied, but also because it is immensely enriched.
...
But what really enriches and enlivens things is that we deal with lots of sigma-algebras, not just the one sigma-algebra which is the concern of measure theory.

WILLIAMS, page xi.

Of course, intuition is much more important than knowledge of measure theory.

WILLIAMS, page xii.

♦ signifies something important, ♦♦ something very important, and ♦♦♦ the Martingale Convergence Theorem.

WILLIAMS, page xiv.

Some questions to ask

Some questions not to ask