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:
“As to the rout that is made about people who are ruined by extravagance, it is no matter to the nation that some individuals suffer. When so much general productive exertion is the consequence of luxury, the nation does not care though there are debtors in gaol; nay, they would not care though their creditors were there too.”
—Samuel Johnson (1709–1784)
“I thought that a Jewish state would be free of the evils afflicting other societies: theft, murder, prostitution.... But now we have them all. And that’s a thing that cuts to the heart ...”
—Golda Meir (1898–1978)
“Stars scribble on our eyes the frosty sagas,
The gleaming cantos of unvanquished space . . .”
—Hart Crane (1899–1932)