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:
“Every writer is necessarily a criticthat is, each sentence is a skeleton accompanied by enormous activity of rejection; and each selection is governed by general principles concerning truth, force, beauty, and so on.... The critic that is in every fabulist is like the icebergnine-tenths of him is under water.”
—Thornton Wilder (18971975)
“If Los Angeles has been called the capital of crackpots and the metropolis of isms, the native Angeleno can not fairly attribute all of the citys idiosyncrasies to the newcomerat least not so long as he consults the crystal ball for guidance in his business dealings and his wife goes shopping downtown in beach pajamas.”
—For the State of California, U.S. public relief program (1935-1943)
“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 (15641616)