Hitting straight lines by compound Poisson process paths

In a recent article Mallows and Nair (1989,Ann. Inst. Statist. Math.,41, 1–8) determined the probability of intersectionP{X(t)=αt for somet≥0} between a compound Poisson process {X(t), t≥0} and a straight line through the origin. Using four different approaches (direct probabilistic, via differential equations and via Laplace transforms) we extend their results to obtain the probability of intersection between {X(t), t≥0} and arbitrary lines. Also, we display a relationship with the theory of Galton-Watson processes. Additional results concern the intersections with two (or more) parallel lines. Work done in part while these authors were visiting professors at the Indian Statistical Institute, Delhi Centre, New Delhi, 110016, India. This author's investigation was supported in part by the U. S. National Science Foundation Grant No. DMS-8504319. Our coauthor and friend Prem Singh Puri died on August 12, 1989. We dedicate our contribution to this paper to his memory.