Some results and applications of geometric counting processes

Among Mixed Poisson processes, counting processes having geometrically distributed increments can be obtained when the mixing random intensity is exponentially distributed. Dealing with shock models and compound counting models whose shocks and claims occur according to such counting processes, we provide various comparison results and aging properties concerning total claim amounts and random lifetimes. Furthermore, the main characteristic distributions and properties of these processes are recalled and proved through a direct approach, as an alternative to those available in the literature. We also provide closed-form expressions for the first-crossing-time problem through monotone nonincreasing boundaries, and numerical estimates of first-crossing-time densities through other suitable boundaries. Finally, we present several applications in seismology, software reliability and other fields.