Content deleted Content added
Citation bot (talk | contribs) Alter: title, template type. Add: volume, series, chapter-url, chapter, authors 1-1. Removed or converted URL. Removed parameters. Some additions/deletions were parameter name changes. | Use this bot. Report bugs. | Suggested by Dominic3203 | Linked from User:Qwfp/One_month_of_probability_and_statistics_page_views | #UCB_webform_linked 56/132 |
No edit summary |
||
Line 193:
Given a Markov transition matrix and an invariant distribution on the states, a probability measure can be imposed on the set of subshifts. For example, consider the Markov chain given on the left on the states <math>A, B_1, B_2</math>, with invariant distribution <math>\pi = (2/7, 4/7, 1/7)</math>. By ignoring the distinction between <math>B_1, B_2</math>, this space of subshifts is projected on <math>A, B_1, B_2</math> into another space of subshifts on <math>A, B</math>, and this projection also projects the probability measure down to a probability measure on the subshifts on <math>A, B</math>.
The curious thing is that the probability measure on the subshifts on <math>A, B</math> is not created by a Markov chain on <math>A, B</math>, not even multiple orders. Intuitively, this is because if one observes a long sequence of <math>B^n</math>, then one would become increasingly sure that the <math>\Pr(A
Conversely, there exists a space of subshifts on 6 symbols, projected to subshifts on 2 symbols, such that any Markov measure on the smaller subshift has a preimage measure that is not Markov of any order (example 2.6<ref name=":1"/>).
| |||