Geometric Rates of Convergence

Abstract

We have seen in Chapter 11 that a positive recurrent irreducible kernel P on $\mathsf {X}\times \mathscr {X}$ admits a unique invariant probability measure, say $\pi$. If the kernel is, moreover, aperiodic, then the iterates of the kernel $P^n(x,\cdot )$ converge to $\pi$ in total variation distance for $\pi$-almost all $x\in \mathsf {X}$. Using the characterizations of Chapter 14, we will in this chapter establish conditions under which the rate of convergence is geometric in f-norm, i.e., $\lim _{n \rightarrow \infty } \delta ^n \left\| P^n(x,\cdot )-\pi \right\| _{f}=0$ for some $\delta > 1$ and positive measurable function f.