Lectures
It is important that you attend the lectures as we do cover material
which is not in these versions. If you miss a lecture
please obtain the missing lecture from a classmate.
Lecture1
Basic foundations (sigma fields, etc...) .
Main Lecture2 - the lecture as
given in class. Please read this as well as the web document Lecture2
which also includes some Branching Processes' material, which was
covered for the most part in STA347.
Lecture2
More basics, Kolmogoroo O-1 Law, L2 spaces, existence of
E(YlX).
Important: For next week's
lecture ( on Feb 3) please read L2intro,
much
of
which is covered in this week's lecture. We will quickly go
over this and move on to renewal processes.
Please look up
the definition of a norm, metric, metric space, inner product,
Kronecker's lemma, Abel's
theorem/lemma.
Lecture3
Renewal processes, coupling, branching processes.
L2intro
A short review of L2, norms, inner products, etc....
Lecture4
We also covered a proof of the Elementary Renewal Theorem ( see lecture
3), Wald's Equation, stopping times. This posted lecture 4 introduces
Markov Chains and sets the stage for using generating functions to
obtain limiting results.
A Couple of Problems:
1. Show using n-step transistion probabilities, without appealing to
generating functions, that states which communicate are either both
transient or both recurrent.
2. # 1, p106 of Whittle. This is a discrete time renewal process with
lifetime pgf G(z)=[1/(1-qz)]^m , where m=1, 2. Calculate ut=E(
#
of
renewals
at time t) and evaluate the limit as t becomes large. Note: Multiple renewals are
possible in this case.
Lecture
5
Some review and some new material dealing with
generating functions applied to Marov Chains as well as a main MC limit
theorem.
Lecture
6
More on the Markov Chain limit theorem, random walks on the integers,
the Ballot problem, gambler's ruin, ...
Lecture
7
Introduction to cts time processes, including the KBE and KFE equations
and a non-rigorous derivation of them ( we will discuss a rigorous
version later).. In addition we also went over the random walk on the
integers with general jump sizes of mean 0 ( please see Lecture 6).
Please read the following for
March 10.
T is finite -this
completes the coupling argument.
Uniform Integrability - useful when
discussing martingales.
Lecture
8
A bit more on cts time processes, the T is finite proof and the short
discusion on uniform integrability.
Lecture9 - discrete time
martingales.
The test will cover the first 9
lectures. Please note that only the first 10 pages of lecture 9 are covered.
Lecture10a - we went over this in detail.
Lecture10b - we looked at some of this. We will continue withh this next week.