Loud shot noise
Let \(\{\) X(t), \(t\geq 0\}\) be a shot noise process of the form \[ X(t)=\sum_{\tau_ i\leq t}h(t-\tau_ i), \] where the response function h is non-negative and non-increasing, and where \(\{\tau_ i\), \(i\geq 0\}\) are renewal times. Define the stationary sequence \[ \xi_ n=\sum_{m\leq n}h(\tau_ n-\tau_ m). \] The authors consider the tail behaviour of the distribution of \(\xi_ 0\), i.e. for what values of \(\delta\) is \(P\{\xi_ 0>x+\delta | \xi_ 0>x\}\to 0\) as \(x\to \infty\). Next they consider the relation \[ P\{M_ n\leq u_ n\}-P^ n\{\xi_ 0\leq u_ n\}\to 0\quad as\quad n\to \infty, \] where \(M_ n=\max_{0\leq i\leq n}\xi_ i\) and lim sup nP\(\{\xi\) \({}_ 0>u_ n\}<\infty\). The results considerably generalize those of \textit{T.-L. Hsing} and the reviewer [Adv. Appl. Probab. 21, No.3, 513-525 (1989; Zbl 0681.60110)].
scientific article; zbMATH DE number 4194907