On crossing times for multidimensional walks with skip-free components

Abstract

The paper deals with processes in(or), one of whose components is skip-free. We obtain identities for distributions of hitting times for the components of the process generalizing the well-known one for the one-dimensional case. These relations reflect the fact that in this case spatial and time coordinates play, in some sense, symmetric roles. They turn out to be useful for solving several problems. For example, they allow us to find the distribution of the number of jumps of the process, which fall in a fixed set before the skip-free component of the process hits a fixed level. Examples are given showing how our results can be applied to models in branching processes, queueing, and risk th..