Lecture 1 Notes: Metric spaces and Topology
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Metric spaces. Open, closed and compact sets
When we discuss probability theory of random processes, the underlying sample
spaces and -eld structures beco
Lecture 2 Notes: Large Deviations for i.i.d. Random Variables
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Preliminary notes
The Weak Law of Large Numbers tells us that if X1 , X2 , . . . , is an i.i.d. sequence of random variables with mean E
Lecture 6 Notes: Brownian motion
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Historical notes
1765 Jan Ingenhousz observations of carbon dust in alcohol.
1828 Robert Brown observed that pollen grains suspended in water perform a continual s
Lecture 9 Notes: Conditional Expectations
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Conditional Expectations
1.1
Denition
Recall how we dene conditional expectations. Given a random variable X
and an event A we dene E[X|A] = E[X1cfw_A] .
Al
Lecture 10 Notes: Martingales
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Martingales
We continue with studying examples of martingales.
Brownian motion. A standard Brownian motion B(t) is a martingale
on C[0, ), equipped with the Wiener mea
Lecture 8 Notes: Quadratic Variation Property
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Unbounded variation of a Brownian motion
Any sequence of values 0 < t0 < t1 < < tn < T is called a partition =
(t0 , . . . , tn ) of an interval [0, T ]
Lecture 3 Notes: Large deviations Theory. Cramers Theorem
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Cramers Theorem
We have established in the previous lecture that under some assumptions on
the Moment Generating Function (MGF) M (), an i.i
Lecture 5 Notes: Extension of LD to Rd and dependent Process
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Large Deviations in Rd
Most of the developments in this lecture follows Dembo and Zeitouni book [1].
Let Xn Rd be i.i.d. random variables
Lecture 4 Notes: Applications of the large deviation technique
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Safety capital for an insurance company
Consider some insurance company which needs to decide on the amount of capital S0 it needs to h
Lecture 7 Notes: The Reection Principle
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Technical preliminary: stopping times
Stopping times are loosely speaking rules by which we interrupt the process
without looking at the process after it was