On the distribution of the supremumof a random walk when the characteristic equation has roots

We consider the random walk $\{S_{n}\}$, generated by a sequence $\{X_{k}\}$ of independent identically distributed random variables with ${\mathbf{E}}X_{1}\in (-\infty,0)$. The influence of the roots of the characteristic equation $1-{\mathbf{E}}\exp(sX_{1})=0$ in the analyticity strip of the Laplace transform ${\mathbf{E}}\exp(sX_{1})$ on the distribution of the supremum $\sup_{n\ge 0}S_{n}$ is studied. An analogous problem is investigated for the stationary distribution of an oscillating random walk.