On the Moment Distance Between Sensors and Anchor Points

TL;DR

This paper extends previous work by deriving an asymptotic expression for the expected moment distance between randomly placed sensors and their ideal positions on a unit interval, revealing how this distance scales with the number of sensors.

Contribution

It provides a new asymptotic formula for the sum of expected moments of sensor distances, specifically for odd natural moment orders, building on earlier results.

Findings

Asymptotic formula for expected moment distances derived

Expected distance scales as n^{-(a/2 - 1)} for odd a

Extends previous results with higher-order asymptotics

Abstract

The present paper contains additional asymptotic result over an earlier investigation of Kapelko and Kranakis. Consider $n$ mobile sensors placed independently at random with the uniform distribution on the unit interval $[0,1]$. Fix $a$ an odd natural number. Let $X_i$ be the the $i-$th closest sensor to $0$ on the interval $[0,1].$ Then the following identity holds $$\sum_{i=1}^n\mathbf{E}\left[\left|X_i-\left(\frac{i}{n}-\frac{1}{2n}\right)\right|^a\right]=\frac{\Gamma\left(\frac{a}{2}+1\right)}{2^{\frac{a}{2}}(1+a)}\frac{1}{n^{\frac{a}{2}-1}}+O\left(\frac{1}{n^{\frac{a-1}{2}}}\right),$$ when $a$ is an odd natural number, where $\Gamma(z)$ is the Gamma function.