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Shot noise distributions and selfdecomposability. (English) Zbl 1042.60026

Summary: Stationary (limiting) distributions of shot noise processes, with exponential response functions, form a large subclass of positive selfdecomposable distributions that we illustrate by many examples. These shot noise distributions are described among selfdecomposable ones via the regular variation at zero of their distribution functions. However, slow variation at the origin of (an absolutely continuous) distribution function is incompatible with selfdecomposability and this is shown in three examples.

MSC:

60G51 Processes with independent increments; Lévy processes
60G07 General theory of stochastic processes
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60K05 Renewal theory
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