Loud Shot Noise

Abstract

We consider problems involving large or loud values of the shot noise process $X(t) := \sum_{i: \tau_i \leq t} h(t - \tau_i), t \geq 0$, where $h: \lbrack 0, \infty) \rightarrow \lbrack 0, \infty)$ is nonincreasing and $(\tau_i, i \geq 0)$ is the sequence of renewal times of a renewal process. Results are obtained by extending the renewal sequence to all $i \in \mathbb{Z}$ and considering the stationary sequence $(\xi_n)$ given by $\xi_n = \sum_{i \leq n} h(\tau_n - \tau_i)$. We show that $\xi_n$ has a thin tail in the sense that under broad circumstances $\operatorname{Pr}\{\xi_n > x + \delta \mid \xi_n > x\} \rightarrow 0$ as $x \rightarrow \infty$, where $\delta > 0$. We also show that $\operatorname{Pr}\{\max(\xi_1, \cdots, \xi_n) \leq u_n\} - (\operatorname{Pr}\{\xi_0 \leq u_n\})^n \rightarrow 0$ for real sequences $(u_n)$ for which $\lim \sup n \operatorname{Pr}\{\xi_0 > u_n\} < \infty$.