## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 1, Number 1 (1991), 88-103.

### Loud Shot Noise

R. A. Doney and George L. O'Brien

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

#### Article information

**Source**

Ann. Appl. Probab., Volume 1, Number 1 (1991), 88-103.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005982

**Digital Object Identifier**

doi:10.1214/aoap/1177005982

**Mathematical Reviews number (MathSciNet)**

MR1097465

**Zentralblatt MATH identifier**

0724.60105

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K99: None of the above, but in this section

Secondary: 60K05: Renewal theory 60J05: Discrete-time Markov processes on general state spaces

**Keywords**

Shot noise extreme values renewal processes

#### Citation

Doney, R. A.; O'Brien, George L. Loud Shot Noise. Ann. Appl. Probab. 1 (1991), no. 1, 88--103. doi:10.1214/aoap/1177005982. https://projecteuclid.org/euclid.aoap/1177005982