  Across a wide variety of applications, the self-exciting Hawkes process has
been used to model phenomena in which the history of events influences future
occurrences. However, there may be many situations in which the past events
only influence the future as long as they remain active. For example, a person
spreads a contagious disease only as long as they are contagious. In this
paper, we define a novel generalization of the Hawkes process that we call the
ephemerally self-exciting process. In this new stochastic process, the
excitement from one arrival lasts for a randomly drawn activity duration, hence
the ephemerality. Our study includes exploration of the process itself as well
as connections to well-known stochastic models such as branching processes,
random walks, epidemics, preferential attachment, and Bayesian mixture models.
Furthermore, we prove a batch scaling construction of general, marked Hawkes
processes from a general ephemerally self-exciting model, and this novel limit
theorem both provides insight into the Hawkes process and motivates the model
contained herein as an attractive self-exciting process in its own right.
