Supremum asymptotics for random walk with switching

We consider random walk $X_n, n\geq 0$ with one level of switching $a\leq 0.$ Some theorems on the asymptotics of the supremum distribution $\mathbf{P}\left(\sup\limits_{n} X_n>x\right)$ as $x\to\infty,$ $a\to -\infty$ were obtained in Cramer case.