## Taiwanese Journal of Mathematics

- Taiwanese J. Math.
- Advance publication (2020), 30 pages.

### On Binomial and Poisson Sums Arising from the Displacement of Randomly Placed Sensors

Michael Fuchs, Louis Kao, and Wan-Zhen Wu

#### Abstract

We re-visit the asymptotics of a binomial and a Poisson sum which arose as (average) displacement costs when moving randomly placed sensors to anchor positions. The first-order asymptotics of these sums were derived in several stages in a series of recent papers. In this paper, we give a unified approach based on the classical Laplace method with which one can also derive more terms in the asymptotic expansions. Moreover, in a special case, full asymptotic expansions can be given which even hold as identities. This will be proved by a combinatorial approach and systematic ways of computing all coefficients of these identities will be discussed as well.

#### Article information

**Source**

Taiwanese J. Math., Advance publication (2020), 30 pages.

**Dates**

First available in Project Euclid: 28 May 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.twjm/1590652820

**Digital Object Identifier**

doi:10.11650/tjm/200503

**Subjects**

Primary: 05A16: Asymptotic enumeration 60C05: Combinatorial probability 68W40: Analysis of algorithms [See also 68Q25]

**Keywords**

sensor displacement cost asymptotics Laplace method generating functions

#### Citation

Fuchs, Michael; Kao, Louis; Wu, Wan-Zhen. On Binomial and Poisson Sums Arising from the Displacement of Randomly Placed Sensors. Taiwanese J. Math., advance publication, 28 May 2020. doi:10.11650/tjm/200503. https://projecteuclid.org/euclid.twjm/1590652820