Harmonic Functions of Random Walks in a Semigroup via Ladder Heights

We investigate harmonic functions and the convergence of the sequence of ratios $$({\mathbb {P}}_x(\tau _\vartheta> n)/{\mathbb {P}}_e(\tau _\vartheta > n))$$ ( P x ( τ ϑ > n ) / P e ( τ ϑ > n ) ) for a random walk on a countable group killed upon the time $$\tau _\vartheta $$ τ ϑ of the first exit from some semigroup 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 multi-dimensional random walks (X(t)) killed upon the times of first exit from a convex cone. Our approach combines large deviation estimates and an extension of Choquet–Deny theory.