Tail probabilities for non-standard risk and queueing processes with subexponential jumps

A well-known result on the distribution tail of the maximum of a random walk with heavy-tailed increments is extended to more general stochastic processes. Results are given in different settings, involving, for example, stationary increments and regeneration. Several examples and counterexamples illustrate that the conditions of the theorems can easily be verified in practice and are in part necessary. The examples include superimposed renewal processes, Markovian arrival processes, semi-Markov input and Cox processes with piecewise constant intensities.