Law of large numbers for a two-dimensional class cover problem

¹ Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849, USA.
² Department of Applied Mathematics and Statistics, The Johns Hopkins University, Baltimore, MD 21218, USA.
³ China Securities, 188 Chao Nei St., Dong Cheng District, Beijing 10010, P.R. China.

^* Corresponding author: ceyhan@auburn.edu

Received: 9 January 2019
Accepted: 29 June 2021

Abstract

We prove a Law of Large Numbers (LLN) for the domination number of class cover catch digraphs (CCCD) generated by random points in two (or higher) dimensions. DeVinney and Wierman (2002) proved the Strong Law of Large Numbers (SLLN) for the uniform distribution in one dimension, and Wierman and Xiang (2008) extended the SLLN to the case of general distributions in one dimension. In this article, using subadditive processes, we prove a SLLN result for the domination number generated by Poisson points in ℝ². From this we obtain a Weak Law of Large Numbers (WLLN) for the domination number generated by random points in [0, 1]² from uniform distribution first, and then extend these result to the case of bounded continuous distributions. We also extend the results to higher dimensions. The domination number of CCCDs and related digraphs have applications in statistical pattern classification and spatial data analysis.