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:
“A point has been reached where the peoples of the Americas must take cognizance of growing ill-will, of marked trends toward aggression, of increasing armaments, of shortening tempers—a situation which has in it many of the elements that lead to the tragedy of general war.... Peace is threatened by those who seek selfish power.”
—Franklin D. Roosevelt (1882–1945)
“A Church which has lost its memory is in a sad state of senility.”
—Henry Chadwick (b. 1920)
“Shall we now
Contaminate our fingers with base bribes,
And sell the mighty space of our large honors
For so much trash as may be grasped thus?
I had rather be a dog and bay the moon
Than such a Roman.”
—William Shakespeare (1564–1616)