Limiting behavior of a random variable conditional on a rare event
K. Zomerdijk (TU Delft - Electrical Engineering, Mathematics and Computer Science)
RICHARD C. KRAAIJ – Mentor (TU Delft - Applied Probability)
Y. van Gennip – Graduation committee member (TU Delft - Mathematical Physics)
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Abstract
As an insurer you want identify the risks you take to prevent bankruptcy. The
theory of large deviations formalizes the study of such rare events. We will use the
theorem of Cramér, which is a main theorem in large deviation theory, to investigate
the rate at which the probability of large deviations of the sums of random variables
decay. Using Sanov’s theorem we will derive an expression for large deviations of
the empirical measure. Furthermore, we will use Gibbs’s principle to derive the
distribution of random variables conditional on a large deviation.