AS100 (Large Deviations)
Fall 2009, Block 1
Course Description: The study of rare events occupies an important part in many probabilistic
analyses. While there are many types of rare events, the theory of
exponentially small events deserves a special consideration. In this course we
will develop the mathematical background for exploring the subject of large
deviations, which studies the structure and transformations of such
exponentially small events. We will start with a study of how quickly the law
of large numbers holds, then develop some general theories, such as the
Gartner-Ellis theorem, the contraction principle and Varadhan's lemma, and then
apply them in a number of ways. For example, we will give precise estimates of
large deviations for a sequence of independent random variables (Cramer's
theorem). If time permits, we will also study rare events that depend on random
processes, e.g., brownian motion. Along the way, we describe how entropy plays
a central role in the calculation of probabilities of large deviations. A more
concrete approach to the subject, based on applications to finance and
actuarial science will be given the corresponding course 'Topics in large
deviations and finance' in Block 3.
Lectures and Exercise Sessions with Magdalena Musat (musat@math.ku.dk)
: Mondays 15-17 (Aud. 7) and Wednesdays 9-10 (Room 109), 10-12 (Aud. 3).
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