Some Properties of the Supremum of Sums of Stationary Related Random Variables

Let $\{\xi_j,\ -\infty<j<\infty\}$ be a strong-sense stationary sequence
$$ X_k=\sum_{j=1}^k \xi_j,\quad X_0=0,\quad \eta=\sup_{k\ge 0}X_k,\quad \theta=\inf_{k\ge 0}X_k. $$

We prove two theorems; the first explains the connection between the nature of $\{\xi_j\}$ and the distributions of $\eta$ and $\theta$; the second gives a useful inequality for $\mathbf{P}(\eta>0)$ in terms of the distribution of $\xi_j$.