Large deviations theory is a branch of probability theory which studies the exponential decay of
probabilities for extremely rare events in the context of sequences of probability distributions. The
theory originates from actuaries studying risk and insurance from a mathematical perspective,
but today it has become its own field of study, and is no longer as tightly linked to insurance
mathematics. Large deviations theory is nowadays frequently applied in various fields, such as
information theory, queuing theory, statistical mechanics and finance. The connection to insurance
mathematics has not grown obsolete, however, and these new results can also be applied to develop
new results in the context of insurance.
This paper is split into two main sections. The first presents some basic concepts from large deviations theory as well as the Gärtner-Ellis theorem, the first main topic of this thesis, and then
provides a fairly detailed proof of this theorem. The Gärtner-Ellis theorem is an important result
in large deviations theory, as it gives upper and lower bounds relating to asymptotic probabilities,
while allowing for some dependence structure in the sequence of random variables.
The second main topic of this thesis is the presentation of two large deviations results developed
by H. Nyrhinen, concerning the random time of ruin as a function of the given starting capital.
This section begins with introducing the specifics of this insurance setting of Nyrhinen’s work as
well as the ruin problem, a central topic of risk theory. Following this are the main results, and the
corresponding proofs, which rely to some part on convex analysis, and also on a continuous version
of the Gärtner-Ellis theorem.
Recommended preliminary knowledge: Probability Theory, Risk Theory.
