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:
“The general feeling was, and for a long time remained, that one had several children in order to keep just a few. As late as the seventeenth century . . . people could not allow themselves to become too attached to something that was regarded as a probable loss. This is the reason for certain remarks which shock our present-day sensibility, such as Montaigne’s observation, “I have lost two or three children in their infancy, not without regret, but without great sorrow.””
—Philippe Ariés (20th century)
“But one day he met a man who was a whole lot badder,
And now he’s dead, and we ain’t none the sadder.”
—Administration in the State of Texa, U.S. public relief program (1935-1943)
“Here were poor streets where faded gentility essayed with scanty space and shipwrecked means to make its last feeble stand, but tax-gatherer and creditor came there as elsewhere, and the poverty that yet faintly struggled was hardly less squalid and manifest than that which had long ago submitted and given up the game.”
—Charles Dickens (1812–1870)