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:
“Even more important than the discovery of Columbus, which we are gathered together to celebrate, is the fact that the general government has just discovered women.”
—Bertha Honore Potter Palmer (1849–1918)
“... the idea of a classless society is ... a disastrous mirage which cannot be maintained without tyranny of the few over the many. It is even more pernicious culturally than politically, not because the monolithic state forces the party line upon its intellectuals and artists, but because it has no social patterns to reflect.”
—Agnes E. Meyer (1887–1970)
“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)