On growth collapse processes with stationary structure and their shot-noise counterparts

In this paper we generalize existing results for the steady state distribution of growth collapse processes. We begin with a stationary setup with some relatively general growth process and observe that under certain expected conditions point and time stationary versions of the processes exist as well as a limiting distribution for these processes which is independent of initial conditions and necessarily has the marginal distribution of the stationary version. We then specialize to the case where an i.i.d. structure holds and specialize to the case where the growth process is a nondecreasing Lévy process and in particular linear and the times between collapses form an i.i.d. sequence. Known results can be seen as special cases for example when the inter-collapse times form a Poisson process or when the collapse ratio is deterministic. Finally we comment on the relation between these processes and shot-noise-type processes and observe that under certain conditions, the steady state distribution of one may be directly inferred from the other.