  We investigate harmonic functions and the convergence of the sequence of
ratios $(P_x(\tau_\vartheta {>} n)/P_e(\tau_\vartheta {>} n))$ for a random
walk on a countable group killed up on the time $\tau_\vartheta$ of the first
exit from some semi-group with an identity element $e$. Several results of
classical renewal theory for one dimensional random walk killed at the first
exit from the positive half-line are extended to a multi-dimensional setting.
For this purpose, an analogue of the ladder height process and the
corresponding renewal function $V$ are introduced. The results are applied to
multidimensional random walks killed upon the times of first exit from a convex
cone. Our approach combines large deviation estimates and an extension of
Choquet-Deny theory.
