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 left undone those things which we ought to have done; and we have done those things which we ought not to have done.”
—Morning Prayer, General Confession, Book of Common Prayer (1662)
“One usually dies because one is alone, or because one has got into something over one’s head. One often dies because one does not have the right alliances, because one is not given support. In Sicily the Mafia kills the servants of the State that the State has not been able to protect.”
—Giovanni Falcone (1939–1992)
“It is not through space that I must seek my dignity, but through the management of my thought. I shall have no more if I possess worlds.”
—Blaise Pascal (1623–1662)