Probability theory

Schedule:

Lectures: Tuesday 15:15-17:00 AAC 137

• First lecture on September10th.
  
• Lecture notes
  

Exercises: Tuesday 17:15-19.00 AAC 137

• First exercise session on September 11'th.
• Questions may also be posted on Moodle

We introduce the notion of a probability space, discussing sigma fields and giving elementary properties of probability measures.

We continue to study probability spaces.  In particular we discuss the

completion and random variables

(and extended random variables).  We then give the celebrated

pi-lambda lemma of Dynkin.

We comolete the unicity part of the definition of product spaces.  Then we begin to defiune the abstract Epectation.

Hopefully we can deal with integral inequalities.

REMEMBER FIRST QUIZ IN EXERCISE SESSION AFTER LECTURE.

We end our treatment of abstract integration with

a discussion of the dominated convergence theorem. 

We then discuss independence: independencce of events, of sigma fields and of

random variables.  We Show that independent pi systems generate indeoendent

sigma fields and give the

useful grouping result for independent random variables.

We finish with independence by treating independence and expectation.  

We discuss different types of convergence for r.v.s: a.s., convergence in probability and convergence L^P.

We consdier basic theorems for convergence i probability for independent r.v.s

concluding with a first weak law of large numbers and the St Petersburg Paradox" exa,mple

We give the law of large numbers.  We then consider series of independent  random variables and, time permitting,

  the 0-1 law.

We continue with BC1 and BC2. We prove the strong law of large numbers

This week we continue to study convergence in distribution, culimating in the

compactness result Helly Borel.

We give the inversion formula for characteristc functions.  We  exploit this to discuss  eg.g.

properties of the Cauchy distribution.  We begin preparations for the CLT

We give the classical characteristic function proof of the central limit theorem and then the more general Lindeber_Feller CLT.

We then give some examples.

We discuss the method of moments and the carleman condition.  We then

give the proof of the local central limit theorem

We show "local central limit" theorems, both for the lattice case and the non latice