The uniform CLT for the empirical estimator of countable state space semi-Markov kernels indexed by functions with applications

This paper is concerned with the uniform CLT for an empirical estimator of countable state space semi-Markov processes indexed by functions under the uniformly integrable entropy condition. The results of this paper generalise those of Limnios [(2004), ‘A Functional Central Limit Theorem for the Empirical Estimator of a Semi-Markov Kernel’, Journal of Nonparametric Statistics, 16(1–2), 13–18. The International Conference on Recent Trends and Directions in Nonparametric Statistics]. We apply our results to the following important topics: on the kernel-type estimator of the semi-Markov kernel, the weak convergence of the smoothed empirical semi-Markov processes, we introduce the copula estimators in the semi-Markov framework and we establish their limiting laws by using the functional delta methods. We prove the main results for the uniform CLT via a martingale difference sequence as in Bae et al. [(2010), ‘The Uniform CLT for Martingale Difference Arrays Under the Uniformly Integrable Entropy’, Bulletin of the Korean Mathematical Society, 47(1), 39–51].