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Asymptotic tail behavior of Poisson shot-noise processes with interdependence between shock and arrival time. (English) Zbl 1321.60104

The authors study the asymtotic tail behavior of the Poisson shot-noise process of the form \[ S\left( t\right) =\sum_{k\geq1}X_{k}h\left( t,\tau_{k}\right) I\left( \tau_{k}\leq t\right) ,\quad t\geq0, \] where \(\left\{ X_{k}:k\geq1\right\} \) is a sequence of identically distributed, but not necessarily independent, nonnegative random variables, the shot function \(h\left( t,s\right) \), nonnegative and Borel measurable, represents the cumulative impact on the system up to time \(t\) due to each shock, and \(\left\{ \tau_{k}:k\geq1\right\}\), independent of \(\left\{ X_{k}:k\geq1\right\} \), is another sequence of nonnegative random variables such that \(N\left( t\right) =\sup\left\{ n:\tau_{n}\leq t\right\} \) is a nonhomogeneous Poisson process. The authors consider the following two situations: (i) the shocks \(\left\{ X_{k}:k\geq1\right\} \) are tail equivalent, upper tail dependent and independent of the arrival process \(\left\{ \tau_{k} :k\geq1\right\} \), and (ii) the shocks \(\left\{ X_{k}:k\geq1\right\} \) are tail equivalent, asymptotically independent and interdependent of the arrival times \(\left\{ \tau_{k}:k\geq1\right\} \). Finally, two examples are presented as illustractions of the main results.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60F99 Limit theorems in probability theory
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