On the number of high excursions of linear growth processes

Uni- and bidirectional linear growth processes arise from a Poisson point process on R × [0, ∞) with intensity measure ℓ × Λ (where ℓ = Lebesgue measure) in a certain way. They can be used to model how “growth” is initiated at random times from random points on the line and from each point proceeds at constant speed in one or both directions. Each finite interval will be completely “overgrown” after an a.s. finite time, a time which equals the maximum of a linear growth process on the interval. We give here, using the coupling version of the Stein Chen method, an upper bound for the total variation distance between the distribution of the number of excursions above a threshold z for a linear growth process in an interval of finite length L, and a Poisson distribution. The bound tends to 0 as L and z grow to x in a proper fashion. The general results are then applied to two specific examples.