# On the expected moments between two identical random processes with application to sensor network

## TL;DR

This paper derives a closed-form formula for the expected moments between two identical random processes and applies it to analyze the transportation cost in sensor network matchings, revealing asymptotic behaviors.

## Contribution

It provides a novel analytical formula for expected moments between identical processes and applies it to optimize transportation costs in sensor networks.

## Key findings

- Expected distance moments are characterized analytically for even powers.
- Transportation cost scales as Θ(n^{b/2+1}) for b ≥ 2.
- Transportation cost scales as O(n^{b/2+1}) for 0 < b < 2.

## Abstract

We give a closed analytical formula for expected distance to the power $a$ between two identical general random processes, when $a$ is an even positive number. As an application to sensor network we prove that the optimal transportation cost to the power $b>0$ of the maximal random bicolored matching with edges $\{X_k,Y_k\}$ is in $\frac{\Theta\left(n^{\frac{b}{2}+1}\right)}{{\lambda}^b}$ when $b \ge 2,$ and in $\frac{O\left(n^{\frac{b}{2}+1}\right)}{{\lambda}^b}$ when $0< b < 2.$