General State Space
Many results for Markov chains with finite state space can be generalized to chains with uncountable state space through Harris chains. The main idea is to see if there is a point in the state space that the chain hits with probability one. Generally, it is not true for continuous state space, however, we can define sets A and B along with a positive number ε and a probability measure ρ, such that
Then we could collapse the sets into an auxiliary point α, and a recurrent Harris chain can be modified to contain α. Lastly, the collection of Harris chains is a comfortable level of generality, which is broad enough to contain a large number of interesting examples, yet restrictive enough to allow for a rich theory.
Read more about this topic: Markov Chain
Famous quotes containing the words general, state and/or space:
“We have seen over and over that white male historians in general have tended to dismiss any history they didn’t themselves write, on the grounds that it is unserious, unscholarly, a fad, too “political,” “merely” oral and thus unreliable.”
—Adrienne Rich (b. 1929)
“Science is neither a single tradition, nor the best tradition there is, except for people who have become accustomed to its presence, its benefits and its disadvantages. In a democracy it should be separated from the state just as churches are now separated from the state.”
—Paul Feyerabend (1924–1994)
“Oh, my. I’d forgotten how much I hate space travel.”
—George Lucas (b. 1944)