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February, 1991 Loud Shot Noise
R. A. Doney, George L. O'Brien
Ann. Appl. Probab. 1(1): 88-103 (February, 1991). DOI: 10.1214/aoap/1177005982


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 $\lim \sup n \operatorname{Pr}\{\xi_0 > u_n\} < \infty$.


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R. A. Doney. George L. O'Brien. "Loud Shot Noise." Ann. Appl. Probab. 1 (1) 88 - 103, February, 1991.


Published: February, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0724.60105
MathSciNet: MR1097465
Digital Object Identifier: 10.1214/aoap/1177005982

Primary: 60K99
Secondary: 60J05 , 60K05

Keywords: Extreme values , renewal processes , Shot noise

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.1 • No. 1 • February, 1991
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