8. Semi-Markov Processes with General State Spaces and Distributional Atoms

This chapter delves into the intricate world of semi-Markov processes with general state spaces and distributional atoms, focusing on ergodic theorems and their convergence rates. The text begins by introducing regenerative semi-Markov processes with D-distributional atoms and regeneration moments, providing essential lemmas and relations that express the distribution functions of regeneration time and transition period duration. It then explores necessary and sufficient conditions for the finiteness and upper bounds of power and exponential moments of these times, using test functions to formulate these conditions. The chapter also presents variants of defining relations for stationary distributions and ergodic theorems with explicit upper bounds for convergence rates. Additionally, it discusses the weighting absolute continuity condition for the distribution function of regeneration time, offering simpler conditions based on one-step transition probabilities. The chapter concludes with a summary of the key findings and their implications for semi-Markov processes with distributional atoms, highlighting the non-overlapping nature of ergodic theorems for these processes and those with atoms. Readers will gain a deeper understanding of the complex interplay between regeneration times, transition periods, and stationary distributions in semi-Markov processes, as well as the practical applications of these theoretical insights.