Let Y1, Y2, · ·· be a stochastic process and M a positive real number. Define TM = inf{n | Yn > M} (TM = + ∞ if for n = 1, 2, ···)· We are interested in the probabilities P(TM <∞) and in particular in the case when these tend to zero exponentially fast when M tends to infinity. The techniques of large deviations theory are used to obtain conditions for this and to find out the rate of convergence. The main hypotheses required are given in terms of the generating functions associated with the process (Yn).

The approximation of sums of independent random variables by compound Poisson distributions with respect to stop-loss distances is investigated. These distances are motivated by risk-theoretic considerations. In contrast to the usual construction of approximating compound Poisson distributions, the method suggested in this paper is to fit several moments. For two moments, this can be achieved by scale transformations. It is shown that the new approximations are more stable and improve the usual approximations by accompanying laws in examples where the probability 1 – pi that the ith summand is zero is not too large.

Let be i.i.d. non-negative random variables with d.f. F and Laplace transform L. Let N be integer valued and independent of In many applications one knows that for y → ∞ and a function φ where in turn τ is the solution of an equation On the basis of a sample of size n we derive an estimator τ n for τ by solving ψ (τ n, Ln(τ n), L′n(τ n), · ··) = 0 where Ln is the empirical version of L. This estimator is then used to derive the asymptotic behaviour of φ (y, τ n, Ln(τ n), L′n(τ n), · ··). We include five examples, some of which are taken from insurance mathematics.