The supremuim of a negative drift random walk with dependent heavy-tailed steps

Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk. When the stationary process is heavy-tailed, the corresponding ruin probabilities are high and the stationary distributions are heavy-tailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke. We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example, by a linear process.