Large deviations of heavy-tailed random sums with applications in insurance and finance

We prove large deviation results for the random sum S(t) = ∑^N(t)_i=1 X_i, t ≧ 0, where (N(t))_t≧0 are non-negative integer-valued random variables and (X_n)_n∈ℕ are i.i.d. non-negative random variables with common distribution function F, independent of (N(t))_t≧0. Special attention is paid to the compound Poisson process and its ramifications. The right tail of the distribution function F is supposed to be of Pareto type (regularly or extended regularly varying). The large deviation results are applied to certain problems in insurance and finance which are related to large claims.