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.