I am working my way through Chapter 4 of
Economic Dynamics: Theory and Computation
and wish to pose the following question related to theorem 4.3.18 on page 90:
Theorem 4.3.18: Let p be a stochastic kernel on some metric space S with markov operator M. The following statements are equivalent:
- The dynamical system (P(S), M) is globally stable (note that P(S) is the set of probability distribution functions defined over S).
- There exists a natural number t such that the Dobrushin coefficient of the t th iterate of p is greater than zero.
Stachurski suggests a more intuitive phrasing of the above theorem: suppose we run two Markov chains from two different starting points x and x'. The dynamical system is globally stable if and only if there is a positive probability that the two chains will meet. To me this sounds suspiciously similar to the definition of an ergodic dynamical system
. However, am I correct to make this connection? Is a dynamical system that has a positive Dobrushin coefficient necessarily ergodic?
Yes, a positive Dobrushin coefficient implies ergodicity.
ReplyDeleteProof: By contradiction. Suppose that for any pair of initial conditions x, x' there is a positive probability of the chains meeting, yet the process is not ergodic. Recall that a process is ergodic if, and only if, all invariant sets have either probability 0 or probability 1, so non-ergodicity means that there are at least two invariant sets whose probability is strictly between 0 and 1. Put x in one of these sets and x' in the other; because the sets are invariant, the trajectories of x and x' can never meet, but this contradicts our initial assumption. Thus, there can be no such invariant sets, and the process is ergodic.
(For much, much more along these lines, see e.g. Lindvall's Lectures on the Coupling Method.)
Cosma,
ReplyDeleteThanks for the pointer! I will definitely check out the book once I have finished Stachurski's book...