Limit laws for k-coverage of paths by a Markov-Poisson-Boolean model

TL;DR

This paper establishes limit laws for the k-coverage of regions and moving particles within a Markov-Poisson-Boolean model, combining stochastic geometry and Markov processes to analyze coverage dynamics over time.

Contribution

It introduces new limit laws for k-coverage in a Markov-Poisson-Boolean model, extending coverage analysis to moving particles along paths.

Findings

Limit laws for static k-coverage at arbitrary times.

Limit laws for k-coverage along a moving particle's path.

Application of stochastic geometry and Markov processes to coverage analysis.

Abstract

Let P := {X_i,i >= 1} be a stationary Poisson point process in R^d, {C_i,i >= 1} be a sequence of i.i.d. random sets in R^d, and {Y_i^t; t \geq 0, i >= 1} be i.i.d. {0,1}-valued continuous time stationary Markov chains. We define the Markov-Poisson-Boolean model C_t := {Y_i^t(X_i + C_i), i >= 1}. C_t represents the coverage process at time t. We first obtain limit laws for k-coverage of an area at an arbitrary instant. We then obtain the limit laws for the k-coverage seen by a particle as it moves along a one-dimensional path.