Asymptotics for random walks with dependent heavy-tailed increments

We consider a random walk $\{S_n\}$ with dependent heavy-tailed increments and negative drift. We study the asymptotics for the tail probability $\mathbf{P}\{\sup\limits_n S_n>x\}$ as $x\to\infty$. If the increments of $\{S_n\}$ are independent then the exact asymptotic behavior of $\mathbf{P}\{\sup\limits_n S_n>x\}$ is well known. We investigate the case in which the increments are given as a one-sided asymptotically stationary linear process. The tail behavior of $\sup\limits_n S_n$ turns out to depend heavily on the coefficients of this linear process.