  Poisson shot noise processes are natural generalizations of compound Poisson
processes that have been widely applied in insurance, neuroscience, seismology,
computer science and epidemiology. In this paper we study sharp deviations,
fluctuations and the stable probability approximation of Poisson shot noise
processes. Our achievements extend, improve and complement existing results in
the literature. We apply the theoretical results to Poisson cluster point
processes, including generalized linear Hawkes processes, and risk processes
with delayed claims. Many examples are discussed in detail.
