Geometric Recurrence and Regularity
We have already seen that successive visits to petite sets play a crucial role in the study of the stability of an irreducible Markov chain. In Chapter 11, the existence of an invariant measure and its expression were obtained in terms of the return time to an accessible petite set. In this chapter, we will begin the study of rates of convergence to an invariant distribution by means of modulated moments of the return time to a petite set. However, in practice, it is with few exceptions difficult to compute these modulated moments. In this chapter, we introduce drift conditions that involve only the kernel P or one of its iterates \(P^n\) rather than the return or hitting times and relate them to the modulated moments of the excursions outside a petite set C. We first consider geometric moments and geometric drift conditions. The corresponding rates of convergence will be obtained in Chapter 15.