## Bernoulli

- Bernoulli
- Volume 24, Number 3 (2018), 1973-1994.

### When do wireless network signals appear Poisson?

H. Paul Keeler, Nathan Ross, and Aihua Xia

#### Abstract

We consider the point process of signal strengths from transmitters in a wireless network observed from a fixed position under models with general signal path loss and random propagation effects. We show via coupling arguments that under general conditions this point process of signal strengths can be well-approximated by an inhomogeneous Poisson or a Cox point processes on the positive real line. We also provide some bounds on the total variation distance between the laws of these point processes and both Poisson and Cox point processes. Under appropriate conditions, these results support the use of a spatial Poisson point process for the underlying positioning of transmitters in models of wireless networks, even if in reality the positioning does not appear Poisson. We apply the results to a number of models with popular choices for positioning of transmitters, path loss functions, and distributions of propagation effects.

#### Article information

**Source**

Bernoulli, Volume 24, Number 3 (2018), 1973-1994.

**Dates**

Received: September 2015

Revised: August 2016

First available in Project Euclid: 2 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1517540465

**Digital Object Identifier**

doi:10.3150/16-BEJ917

**Mathematical Reviews number (MathSciNet)**

MR3757520

**Zentralblatt MATH identifier**

06839257

**Keywords**

propagation loss rate of convergence signal strengths total variation distance weak convergence

#### Citation

Keeler, H. Paul; Ross, Nathan; Xia, Aihua. When do wireless network signals appear Poisson?. Bernoulli 24 (2018), no. 3, 1973--1994. doi:10.3150/16-BEJ917. https://projecteuclid.org/euclid.bj/1517540465