Graduate Course in  WEAK CONVERGENCE OF PROBABILITY MEASURES  (minimum 6 Credits)

Starting at the beginning of February, I will give a course on this (--||--), following  

PATRICK BILLINGSLEY: CONVERGENCE OF PROBABILITY MEASURES, Second Edition, Wiley 1999. 

The first edition from 1968 is one of the great CLASSIC TEXTs of probability.

The course will be preceeded by a TWO WEEK INTRODUCTION, starting the latter part of January, 
about TOPOLOGY FOR PROBABILISTS (2 Credits). Topology deals with "convergence, continuity and 
open and closed sets", and such matters. If there is any math probabilists should know about, 
other than abstract Lebesgue integration, it is this. (What is a Borel set in non-Euclidian 
space?) Specifically, topology is needed to get into weak convergence. Also, topology is the 
most easily accessible piece of modern math, and as such, the best of exercise grounds for 
improving in proving things and understanding theory. For the topology introduction, I plan 
to use some lecture notes by myself. (The classic introductory text is the first part of

Simmons: Introduction to Topology and Modern Analysis. McGraw-Hill 1963,

which is a 5 credit graduate course, and is strongly recommended for anyone who have time for 
that (you will never regret it)). Students who have taken a topology course before can omit 
the introdution, and will obviously not get credits for topology once more.

I plan to complete Billingsley with some additional material on, for example,

    general characteristic functions (functionals) - how they work in more general contexts 
    than Euclidian R^n, e.g., to get convergence of continuous stochastic processes from that
    of the corresponding characteristic functions (of the whole processes); 

    general weak convergence (weak convergence of probability measures is a special case of
    general weak convergence (=pointwise convergence) - this will be somewhat looked into, 
    since that clerifies matters, and simlifies understanding;

    convergence in some other important particular contexts than those covered in Billingsley 
    (e.g., convergence of point processes and of probabilities on Banach spaces).
  
Instead, some material in latter sections in Billingsley will be omitted. When I had course 
on weak convergence 10 years ago in Lund, I mainly used excerts from the book (do not buy it)

Araujo & Gine: The Central Limit Theorem for Real and Banach Valued Random Variables, 1980

to complete Billingsley. Now I think I will use the following book instead (do not buy it)

Ledoux & Talagrand: Probability in Banach Spaces; Isoperimetry and Processes, Springer 1991.

Weak convergence is part of the basic knowledge that probabilists have to possess, second in 
necessity only to Lebesgue integration. In any article on probability, weak convergence is 
assumed basic, and its tools invoked without dwelling on details, in the same way as are the-
orems from integration theory. Weak convergence is about matters such as convergence of a 
sequence of continuous stochastic processes to a limit. For example, when generating a sampl-
ed approximation of a stochastic process in a computer by simulation, one wants such converg-
ence (as sampling is refined), to the actual process one tries to simulate. This convergence 
does not follow simply by checking convergence of the finite dimensional distributions, and
weak convergence comes into play. (Weak convergence is not at all restricted to stochastic 
processes, but when asking for examples, they are the most natural ones.) To learn about weak 
convergence is one of the best ways to improve ones understanding of multidimensional stocha-
stic phenomena (e.g., stochastic processes), and their analysis, which is required if one
wants to do anything outside the Gaussian (second-order process) world. (My favorite remark.) 

PREREQUISITES for the course are Lebesgue integration, together with BASIC knowledge of PRO-
BABILITY THEORY (preferably, measure theoretic such). In addition, a graduate or undergradu-
ate course on STOCHASTIC PROCESSES (for example TMA 421 "Stokastiska Processer för E", given 
November - mid December)) is very helpful, since most of the many examples of convergence 
that are treated relate to stochastic processes, and things may become too academic if one 
has not seen such before. Students that know that their strenght is not theory, in order to 
feel comfortable, perhaps should have taken ONE MORE THEORETICAL COURSE, other than Lebesgue 
integration (for example "analytiska metoder i sannolikhetsteorin", given November - beginn-
ing of January). However, the material to cover is quite basic, and should be accessible to
anyone wanting to learn about it (not necessarily without a little paine though, but this is 
the way it should be, since one does not develop much from doing familiar things all over 
again). (I think Billingsley once was mandatory for PhD in Mathematical Statistics.)

GRADING will be by exercise sessions, by handing in solutions to exercises, or by "hemtenta".

Around the middle of January, I will send out a "call for (assembly to) introductory meeting" 
at which we will agree on times to be used for teaching (both for the INTRODUCTION and the 
main part of the course). People who want to discuss things about the course and/or their 
participation are wellcome to contact me, e.g., by email, or in person. Everyone is wellcome.

Best regards, Patrik Albin