Asymptotic ruin probabilities for risk processes with dependent increments

In this paper, we derive a Lundberg type result for asymptotic ruin probabilities in the case of a risk process with dependent increments. We only assume that the probability generating functions exist, and that their logarithmic average converges. Under these assumptions we present an elementary proof of the Lundberg limiting result, which only uses simple exponential inequalities, and does not rely on results from large deviation theory. Moreover, we use dependence orderings to investigate, how dependencies between the claims affect the Lundberg coefficient. The results are illustrated by several examples, including Gaussian and AR(1)-processes, and a risk process with adapted premium rules.