Explicit Criteria for Several Types of Ergodicity

The explicit criteria, collected in Tables 5.1 and 5.2, for several types of ergodicity of onedimensional diffusions or birth-death processes have been found out recently in a surprisingly short period. One of the criteria is for exponential ergodicity of birth-death processes. This problem has been opened for a long time in the study of Markov chains. The survey article explains in details the idea which leads to solve the problem just mentioned. It is interesting that the problem is connected with several branches of mathematics. Some open problems for the further study are also proposed. Let us begin with the paper by recalling the three traditional types of ergodicity. 1. Three traditional types of ergodicity. Let Q = (qij) be a regular Q-matrix on a countable set E = {i, j, k, · · · }. That is, qij ≥ 0 for all i 6= j, qi := −qii = ∑ j 6=i qij < ∞ for all i ∈ E and Q determines uniquely a transition probability matrix P (t) = (pij(t)) (which is also called a Q-process or a Markov chain). Denote by π = (πi) a stationary distribution of P (t): πP (t) = π for all t ≥ 0. From now on, assume that the Q-matrix is irreducible and hence the stationary distribution π is unique. Then, the three types of ergodicity are defined respectively as follows. Ordinary ergodicity : lim t→∞ |pij(t)− πj | = 0 (1.1) Exponential ergodicity : lim t→∞ e|pij(t)− πj | = 0 (1.2) Strong ergodicity : lim t→∞ sup i |pij(t)− πj | = 0 ⇐⇒ lim t→∞ e sup i |pij(t)− πj | = 0, (1.3) where α̂ and β̂ are (the largest) positive constants and i, j varies over whole E. The definitions are meaningful for general Markov processes once the pointwise convergence is replaced by the convergence in total variation norm. The three types of ergodicity were studied in a great deal during 1953–1981. Especially, it was proved that strong ergodicity =⇒ exponential ergodicity =⇒ ordinary ergodicity. Refer to Anderson (1991), Chen (1992, Chapter 4) and Meyn and Tweedie (1993) for details and related references. The study is quite complete in the sense that we have the following criteria which are described by the Q-matrix plus a test sequence (yi) only. Theorem 1.1 (Criteria). Let H 6= ∅ be an arbitrary but fixed finite subset of E. Then the following conclusions hold. (1) The process P (t) is ergodic iff the system of inequalities