A SLLN for a one-dimensional class cover problem

Class cover catch digraphs arise in classification problems in statistical pattern recognition. We prove a strong law of large numbers for the domination number in a random one-dimensional model of class cover catch digraphs. The proof avoids complicated computations due to the dependence of random variables by considering a related Poisson process problem where we may apply classical strong law results and Chernoff exponential probability bounds. Complete convergence in the Poisson representation establishes the desired result for the original problem.