Hitting straight lines by compound Poisson process paths (Q806186)

Let (X(t), \(t\geq 0)\) be a compound Poisson process with intensity \(\lambda\) and jump size distribution function H. The authors consider the probability h(\(\beta\)) of intersection between X and arbitrary lines: \(h(\beta) = P(X(t) = \alpha t+\beta\) for some \(t>0)\) if \(\alpha\) is taken fixed. This work was inspired by \textit{C. L. Mallows} and \textit{V. N. Nair} [Ann. Inst. Stat. Math. 41, No.1, 1-8 (1989; Zbl 0692.60034)], who had considered the no intercept case \(\alpha =1\), \(\beta =0\) and who had obtained h(0) in a form which could be related by the present authors to the theory of Galton-Watson branching processes and their probability of extinction. Four different approaches are used in the paper: a probabilistic method to determine h(\(\beta\)) explicitly for \(\beta <0\) and for \(\alpha >1\), \(\beta >0\); a differential equation established for h(\(\beta\)); a Laplace transform \(\hat h\) of h and a random walk for a discrete time version of the problem which could be applied to the continuous case. The discrete time version gave the clue to a Galton- Watson process interpretation. Finally the intersection of the compound Poisson process with two (or even more) parallel lines is considered.